{["T10", "(Ord(S) & Ord(T)) imp (S incin T or T incin S)"], ["T110", "     (X in S & Y in S) imp (EXISTS f in OM | (one_1_map(f) & range(f) = S & domain(f) = S & f~[X] = Y & f~[Y] = X))"], ["T210", "(K in Si & N in Si & M in Si) imp (N S_PLUS (M S_PLUS K) = (N S_PLUS M) S_PLUS K)"], ["TU10", "Ord(t) & P(t) & t incin o & (FORALL x in t | not P(x))"], ["T50", "range(F ON  S) incin range(F)"], ["T150", "N TIMES (M TIMES K) = (N TIMES M) TIMES K"], ["T250", "(X in Si) imp is_nonneg(X S_TIMES  X)"], ["TU50", "(N in Re)  imp (R_Rev(R_Rev(N)) = N)"], ["T90", "#S  incs  #T eq (T = 0 or (EXISTS f in OM | Svm(f) & domain(f) = S & range(f) = T))"], ["T190", "(not Finite(S)) imp (EXISTS T | (#(T PROD {0,1}) = #S))"], ["TU90", "(N in RF & m in RF) imp (N F_PLUS m = m F_PLUS N)"], ["T20", "     (Ord(X) & Ord(W) & X /= W) imp         (S in {enum(y,S): y in X} or S in {enum(y,S): y in W} or enum(X,S) /= enum(W,S))"], ["T120", "Card(Z)"], ["T220", "     (K in Si & N in Z & M in Z) imp ([N,0] S_TIMES ([M,0] S_TIMES  K) = ([N,0] S_TIMES  [M,0]) S_TIMES  K)"], ["TU20", "(FORALL x | ((cdr(x) in s) imp (sigma({x}) = cdr(x))))"], ["T60", "one_1_map(F) imp (F @ inv(F) = ident(range(F)) & inv(F) @ F = ident(domain(F)))"], ["T160", "(N * M = 0 & N2 * M2 = 0 & #N = #N2 & #M = #M2) imp (#(N + M) = #(N2 + M2))"], ["TU60", "(X in Re & Y in Re & X1 in Re & X R_GT Y & X1 R_GT R_0) imp (X R_TIMES X1 R_GT Y R_TIMES X1)"], ["T30", "(X in F) imp (car(X) in domain(F))"], ["T130", "N TIMES M = N TIMES #M"], ["T230", "(K in Si) imp (K S_PLUS [0,0] = K)"], ["TU30", "(N in Re) imp (N = N R_PLUS R_Rev(N) = R_0)"], ["T70", "(ord(S) & X in S) imp (enum(X,S) = X)"], ["T170", "(M in Z & N in Z & K in N) imp (M PLUS K in M PLUS N)"], ["T270", "(M in Ra) imp (M = M Ra_PLUS Ra_0)"], ["TU70", "(N in Cm) imp (N = N C_PLUS C_0)"], ["T100", "     (one_1_map(F) & S incin domain(F) & T incin domain(F) & S /= T) imp (range(F ON  S) /= range(F ON  T))"], ["T200", "(N in Z) imp (Red([N,N]) = [0,0])"], ["T300", "(N in Re & M in Re) imp (N R_PLUS M = M R_PLUS N)"], ["T40", "(U = [A,B]) imp (U = [car(U),cdr(U)])"], ["T140", "#(N PROD {C}) = #N"], ["T240", "Ult_membs({S}) = {S} + Ult_membs(S)"], ["TU40", "(N in Re & M in Re & R_is_nonneg(R_Rev(M))) imp (N R_GT N R_PLUS M or N = N R_PLUS M)"], ["T80", "Card(#S) & (EXISTS f in OM | one_1_map(f) & range(f) = S & domain(f) = #S)"], ["T180", "(#N = 0) imp (N = 0)"], ["T280", "Ra_is_nonneg(Ra_0) & Ra_is_nonneg(Ra_1)"], ["TU80", "(N in Cm & M in Cm & M /= C_0) imp (C_abs(N) R_OVER C_abs(M) = C_abs(N C_OVER M))"], ["T14", "     not (EXISTS ordinals | (FORALL x | ((x in ordinals) eq Ord(x))))"], ["T114", "Finite(S) imp Finite(next(S))"], ["T214", "(N in Z & M in Z) imp (S_Rev(Red([M,N])) = Red([N,M]))"], ["TU14", "R(m,j) & (FORALL k in m | (FORALL j in Z | not R(k,j))) & (FORALL i in j | not R(m,i))"], ["T54", "is_map(F) imp (F = inv(inv(F)))"], ["TU4", "{e(x,y): x  in s, y in f(x) | p(x,y)} = {ep2(x,y): x  in s, y in fp(x) | pp2(x,y)}"], ["T154", "pow(0) = {0}"], ["T254", "(X in Fr & Y in Fr & Same_frac(X,Y) & W in Fr & Z in Fr & Same_frac(W,Z)) imp     Same_frac([(car(X) S_TIMES  cdr(W)) S_PLUS (car(W) S_TIMES  cdr(X)),cdr(X) S_TIMES  cdr(W)],        [(car(Y) S_TIMES  cdr(Z)) S_PLUS (car(Z) S_TIMES  cdr(Y)),cdr(Y) S_TIMES  cdr(Z)])"], ["TU54", "(M in Re) imp (M = M R_TIMES R_1)"], ["T94", "#S in #T or #S = #T or #T in #S"], ["T194", "(not Finite(S)) imp (#(S PROD S) = #S)"], ["T294", "(X in Ra & Y in Ra & X Ra_GT Y) imp         (X Ra_GT (X Ra_PLUS Y) Ra_OVER (Ra_1 + Ra_1) & (X Ra_PLUS Y) Ra_OVER (Ra_1 + Ra_1) Ra_GT Y)"], ["TU94", "(K in RF & n in RF & m in RF) imp (n F_TIMES (m F_PLUS K) = (n F_TIMES m) F_PLUS (n F_TIMES K))"], ["Tproduct_order.4", "(FORALL x in o1 PROD o2 | (FORALL y in o1 PROD o2 |                 ((Ord1p2(x,y) or Ord1p2(y,x) or x = y) & (not Ord1p2(x,x)))))"], ["T24", "S * T = {x in S | x in T}"], ["T124", "Svm({[X,Y]}) & one_1_map({[X,Y]}) & {[X,Y]}~[X] = Y"], ["T224", "(N in Si & M in Si) imp (N = M S_PLUS (N S_MINUS M))"], ["TU24", "     ((FORALL f in OM | (FORALL g | ((Finite(f) & Finite(g) & Svm(f) & Svm(g) & domain(f) = domain(g)) imp             sigma(f) = sigma({[y,sigma(f INV_IM {y})]: y in range(g)})))))"], ["T64", "(Svm(F) & Svm(G) & X in domain(G) & range(G) incin domain(F)) imp ((F @ G)~[X] = F~[G~[X]])"], ["T164", "(FORALL x in S | Ord(x)) imp Ord(Un(S))"], ["T264", "(N in Ra & M in Ra) imp (N Ra_PLUS M = M Ra_PLUS N)"], ["TU64", "(X in Re & R_is_nonneg(X)) imp (sqrt(X) in Re & R_is_nonneg(sqrt(X)) & sqrt(X) R_TIMES sqrt(X) = X)"], ["T34", "F ON  (A + B) = (F ON  A) + (F ON  B)"], ["T134", "Finite(N) & Finite(M) eq Finite(N PLUS M)"], ["T234", "             (FORALL n in Si | (FORALL m in Si | (FORALL k in Si | ((m S_TIMES n = k S_TIMES  n & n /= [0,0]) imp (m = k)))))"], ["TU34", "(N in Re & M in Re) imp (N = M R_PLUS (N R_MINUS M))"], ["T74", "         (Ord(S) & T incin S & X in S & Y in X) imp (enum(Y,T) in enum(X,T) or enum(X,T)  incs  T)"], ["T174", "(M in Z & N in Z & K in Z) imp (M PLUS K incin N PLUS K eq M incin N)"], ["T274", "     (K in Si & N in Si & M in Si & K /= [0,0] & M /= [0,0]) imp (Fr_to_Ra([N,M]) = Fr_to_Ra([K S_TIMES  N,K S_TIMES  M]))"], ["TU74", "(N in Cm & M in Cm) imp (N C_TIMES M = M C_TIMES N)"], ["Tfcn_symbol.4", "Svm(g)"], ["T104", "             (Svm(F) & Finite(domain(F))) imp Finite(range(F))"], ["T204", "(K in Si & N in Z & M in Z) imp (K S_TIMES  [N,M] = K S_TIMES  Red([N,M]))"], ["T4", "arb(arb(arb([X,Y] - {arb([X,Y])}) - {arb([X,Y])})) = Y"], ["T44", "(domain(F) = 0) eq (range(F) = 0)"], ["T144", "N PLUS M = #((N PROD {0}) + (M PROD {1}))"], ["TU44", "(N in Re & M in Re) imp (abs(N) R_PLUS abs(M) R_GT abs(N R_MINUS M) or abs(N) R_PLUS abs(M) = abs(N R_MINUS M))"], ["T84", "one_1_map(F) imp (#range(F) = #domain(F))"], ["T184", "(N in Z & M in Z) imp ((N PLUS M) MINUS M = N)"], ["T284", "(X in Ra & Y in Ra & Ra_is_nonneg(X) & Ra_is_nonneg(Y)) imp                 (Ra_is_nonneg(X Ra_PLUS Y) & Ra_is_nonneg(X Ra_TIMES Y))"], ["TU84", "(K in Cm & n in Cm & m in Cm) imp (n C_TIMES (m C_PLUS K) = (n C_TIMES m) C_PLUS (n C_TIMES K))"], ["Twell_ordered_set.4", "     (R incin s & T incin R & T /= 0) imp (Minrel(R) = Minrel(T) or Ordrel(Minrel(R),Minrel(T)))"], ["T18", "enum(X,S) = S or enum(X,S) in S"], ["T118", "Ord(Z) & (not Finite(Z)) & (FORALL x in OM | (Card(x) & Finite(x)) eq x in Z)"], ["T218", "(N in Si) imp (S_Rev(N) in Si & S_Rev(N) S_PLUS N = [0,0] & S_Rev(S_Rev(N)) = N)"], ["TU18", "h1(X) = if x = 0 then f1 else g2(h(X - {arb(X)}),X) end if"], ["T58", "         one_1_map(ident(S)) & domain(ident(S)) = S & range(ident(S)) = S & inv(ident(S)) = ident(S) &           (FORALL x in S | (ident(S)~[x] = x)) &                 (is_Map(F) imp (((domain(F) incin S) imp (F @ ident(S) = F)) & ((range(F) incin S) imp (ident(S) @ F = F))))"], ["TU8", "     rng = {x: x in s | f(x) = arb({f(y): y in s})} & rng /= 0 & (FORALL x in rng | (FORALL y in s | f(x) incin f(y)))"], ["T158", "N MINUS 0 = #N"], ["T258", "(X in Fr) imp Same_frac(X,[Si_Rev(car(X)),Si_Rev(cdr(X))])"], ["TU58", "(X in Re) imp (X = X R_TIMES R_1)"], ["T98", "(Finite(S) & S  incs  T) imp Finite(T)"], ["T298", "(N in Re) imp (EXISTS m in Ra | (FORALL x in N | (m Ra_GT x)))"], ["T28", "(G incin F & Svm(F)) imp Svm(G)"], ["T128", "N PLUS M = N PLUS #M"], ["T228", "(K in Si & M in Si) imp (K S_MINUS M = K S_PLUS (M S_TIMES  [0,1]))"], ["TU28", "(FORALL x in s | P(x,arb(f(x))))"], ["T68", "(F = {[[[x,y],z],[x,[y,z]]]: x in A, y in B, z in C}) imp             (one_1_map(F) & domain(F) = ((A PROD B) PROD C) & range(F) = (A PROD (B PROD C)))"], ["T268", "(X in Ra & Y in Si & Z in Si & Z /= [0,0]) imp (Fr_to_Ra([Y,Z]) Ra_TIMES X =                                             Fr_to_Ra([car(arb(X)) S_TIMES  Y,cdr(arb(X)) S_TIMES  Z]))"], ["TU68", "(N in Cm & M in Cm) imp (N C_PLUS M in Cm)"], ["T38", "(X in domain(F)) imp (F~[X] in range(F))"], ["T138", "#N PLUS 0 = #N"], ["T238", "(X in S & Y in X) imp (Y in Ult_membs(S))"], ["TU38", "(N in Re) imp (abs(N) in Re & (abs(N) R_GT N or abs(N) = N) & (abs(N) R_GT R_0 or abs(N) = R_0) & R_is_nonneg(abs(N)))"], ["T78", "     (Ord(S) & T incin S) imp (#T incin S)"], ["T178", "(M in Z & N in Z & N /= 0) imp                     (M OVER N in Z & M  incs  ((M OVER N) TIMES N) & M MOD N in N)"], ["T278", "(M in Ra & M /= Ra_0) imp (Recip(M) in Ra & M Ra_TIMES Recip(M) = Ra_1)"], ["TU78", "(N in Cm & M in Cm) imp     ((C_abs(N) C_PLUS C_abs(M)) R_GT  C_abs(N C_PLUS M) or C_abs(N) C_PLUS C_abs(M) = C_abs(N C_MINUS M))"], ["T108", "(Ord(S) & Finite(S) & T in S) imp Finite(T)"], ["T208", "(J in Z & K in Z & N in Z & M in Z) imp ([J,K] S_PLUS [N,M] = Red([J,K]) S_PLUS Red([N,M]))"], ["T8", "(Ord(S) & Ord(T) & T incin S) imp (T = S or T = arb(S - T))"], ["T48", "Svm(F) imp Svm(F ON  S)"], ["T148", "(A PROD X * B PROD X) = (A * B) PROD X  & (A PROD X + B PROD X) = (A + B) PROD X &             (X PROD A * X PROD B) = X PROD (A * B) & (X PROD A + X PROD B) = X PROD (A + B)"], ["T248", "(X in Si) imp ((is_nonneg(X) or is_nonneg(S_Rev(X))) & ((is_nonneg(X) & is_nonneg(S_Rev(X))) imp (X = [0,0])))"], ["TU48", "(N in Re & M in Re) imp (N R_TIMES M in Re)"], ["T88", "#range(F) incin #F"], ["T188", "(N in Z & M in Z) imp (N = M PLUS (N MINUS M) or N = M MINUS (M MINUS N))"], ["TU88", "C_0 in Cm & C_1 in Cm"], ["Twell_ordered_set.8", "     (EXISTS o | Ord(o) & s = {orden(x): x in o} & (FORALL x in o | orden(x) /= s) & one_1_map({[x,orden(x)]: x in o}))"], ["Tiz_map.1", "is_map(f)"], ["T11", "(Ord(S) & Ord(T) ) imp (S in T or T in S or S = T)"], ["T111", "Svm(F) imp (F ON  S = {[x,F~[x]]: x in domain(f) | x in S} & domain(F ON  S) = {x in domain(F) | x in S} &                         range(F ON  S) = {F~[x]: x in domain(f) | x in S})"], ["T211", "     (K in Si & N in Si & M in Si) imp (N S_TIMES  (M S_PLUS K) = (N S_TIMES  M) S_PLUS (N S_TIMES  K))"], ["TU11", "P(m) & m in Ult_membs({n}) & (FORALL k in m | not P(k))"], ["T51", "(range(G) incin domain(F)) imp (range(F @ G) = range(F ON  range(G)) & domain(F @ G) = domain(G))"], ["TU1", "{e(x): x  in s | p(x)} = {ep1(x): x  in s | pp1(x)}"], ["T151", "N TIMES (M PLUS K) = (N TIMES M) PLUS (N TIMES K)"], ["T251", "(X in Si & Y in Si & X /= [0,0] & is_nonneg(X)) imp (is_nonneg(X S_TIMES  Y) eq is_nonneg(Y))"], ["TU51", "(K in Re & n in Re & m in Re) imp (n R_TIMES (m R_TIMES K) = (n R_TIMES m) R_TIMES K)"], ["T91", "#S = #T eq (EXISTS f in OM | one_1_map(f) & domain(f) = S & range(f) = T)"], ["T191", "(Card(S) & (not Finite(S))) imp (#(S PROD {0,1}) = #S)"], ["T291", "(X in Ra & Y in Ra & X1 in Ra & X Ra_GT Y & X1 Ra_GT Ra_0) imp (X Ra_TIMES X1 Ra_GT Y Ra_TIMES X1)"], ["TU91", "(N in RF & m in RF) imp (N F_TIMES m = m F_TIMES N)"], ["Tproduct_order.1", "(FORALL x in o1 PROD o2 | Ord(car(x)))"], ["T21", "(FORALL s | (EXISTS x | (Ord(x) & s in {enum(y,s): y in x})))"], ["T121", "0 in Z & 1 in Z & 2 in Z & 3 in Z & 1 /= 0 & 2 /= 0 & 3 /= 0 & 1 /= 2 & 1 /= 3 & 2 /= 3"], ["T221", "     (K in Si & N in Z & M in Si) imp ([N,0] S_TIMES  (M S_TIMES  K) = ([N,0] S_TIMES  M))"], ["TU21", "(Finite(F) & range(F) incin s) imp (sigma(F) in s)"], ["Ttransfinite_induction.1", "not (FORALL m | (P(m) imp (EXISTS k in m | P(k))))"], ["T61", "     (is_map(F) & is_map(G) & domain(F) = range(G) & range(F) = domain(G) &             F @ G = ident(range(F)) & G @ F = ident(domain(F))) imp (one_1_map(F) & G = inv(F))"], ["T161", "(M incin N) imp (#N = #M PLUS (N MINUS M))"], ["T261", "     (X in Fr & Y in Fr & Same_frac(X,Y)) imp (is_nonneg(car(X) S_TIMES  cdr(X)) eq is_nonneg(car(Y) S_TIMES  cdr(Y)))"], ["TU61", "(X in Re & X R_GT R_0) imp (Recip(X) R_GT R_0)"], ["T31", "(X in F) imp (cdr(X) in range(F))"], ["T131", "(Finite(N) & M incin N & M /= N) imp (#M in #N)"], ["T231", "(K in Si) imp ([0,0] S_PLUS K = K)"], ["T50a", "domain(F ON  S) = domain(F) * S"], ["TU31", "(N in Re & M in Re) imp (N incin M or M incin N)"], ["T71", "Ord(#S) & (EXISTS f in OM | (one_1_map(f) & range(f) = S & domain(f) = #S) &                 (not((EXISTS o in #S | (EXISTS g | one_1_map(g) & range(g) = S & domain(f) = o)))))"], ["T171", "(M in Z & N in Z & K in Z & M PLUS K = N PLUS K) imp (M = N)"], ["T271", "(M in Ra) imp (M Ra_PLUS Ra_Rev(M) = Ra_0)"], ["T64a", "(Svm(F) & Svm(G) & X in domain(G) & range(G) incin domain(F)) imp         ((F @ G)~[X] = F~[G~[X]] & F @ G = {[x,F~[G~[x]]]: x in domain(g)} & range(F @ G) = {F~[G~[x]]: x in domain(G)})"], ["TU71", "(N in Cm) imp (C_Rev(N) in Cm & C_Rev(C_Rev(N)) = N)"], ["Tfcn_symbol.1", "domain(g) = s"], ["T101", "(Svm(F) & Svm(G) & range(F) incin domain(G)) imp     (G @ F = {[x,G~[F~[x]]]: x in domain(F)} & domain(G @ F) = domain(F) & range(G @ F) = {G~[F~[x]]: x in domain(F)})"], ["T201", "(J in Z & K in Z & M in Z) imp (Red([J PLUS M,K PLUS M]) = Red([J,K]))"], ["T301", "(N in Re) imp (N = N R_PLUS R_0)"], ["T1", "arb({X}) = X"], ["T41", "(is_map(F) & U in F) imp (U = [car(U),cdr(U)])"], ["T1a", "arb({{X},X}) = X"], ["T141", "1 TIMES N = #N"], ["T241", "Ord(S) imp (Ult_membs({S}) = S + {S})"], ["T51a", "(range(G) = domain(F)) imp (range(F @ G) = range(F) & domain(F @ G) = domain(G))"], ["TU41", "(N in Re & M in Re & R_is_nonneg(N) & (not R_is_nonneg(M))) imp     (N R_GT abs(N R_PLUS M) or N = abs(N R_PLUS M) or R_Rev(M) R_GT abs(N R_PLUS M) or R_Rev(M) = abs(N R_PLUS M))"], ["T281a", "(X Ra_GE Y eq nneg(X pluz Ra_Rev(Y))) & (X S_LE Y eq Y Ra_GE X) & (X Ra_GT Y eq X Ra_GE Y & X /= Y) &                                                                 (X Ra_LT Y eq Y Ra_GT X)"], ["T81", "#S = 0 eq S = 0"], ["T181", "#N TIMES #M = 0 eq N = 0 or M = 0"], ["T281", "   (X in Ra) imp ((Ra_is_nonneg(X) or Ra_is_nonneg(Ra_Rev(X))) & ((Ra_is_nonneg(X) & Ra_is_nonneg(Ra_Rev(X))) imp (X = Ra_0)))"], ["T85a", "(F incin G) imp (range(F) incin range(G) & domain(F) incin domain(G))"], ["TU81", "(N in Cm & M in Cm) imp (N C_TIMES M in Cm)"], ["Twell_ordered_set.1", "(FORALL t incin s | (EXISTS x | ((t /= 0) imp (x in t & (FORALL y in t | Ordrel(x,y) or x = y)))))"], ["T39a", "Svm(F) imp (F = {[x,F~[x]]: x in domain(F)} & range(F) = {F~[x]: x in domain(F)})"], ["T249a", "(X S_GE Y eq nneg(X pluz S_Rev(Y))) & (X S_LE Y eq Y S_GE X) &                 (X S_GT Y eq X S_GE Y & X /= Y) & (X S_LT Y eq Y S_GT X)"], ["T15", "Ord(S) imp Ord(next(S))"], ["T115", "not(Finite(s_inf))"], ["T215", "(N in Si & M in Si) imp (N S_TIMES  S_Rev(M) = S_Rev(N S_TIMES  M))"], ["TU15", "(FORALL n in Z | (EXISTS h | (FORALL s | (FORALL x | ((#x incin n) imp (h(x,s) = f({g2(h(y,s),s): y incin x | y /= x & P(x,y,s)},x,s)))))))"], ["T55", "     one_1_map(F) imp (one_1_map(inv(F)) & F = inv(inv(F)) & range(inv(F)) = domain(F) & domain(inv(F)) = range(F))"], ["TU5", "Svm(g)"], ["T155", "Finite(N) eq Finite(pow(N))"], ["T255", "(X in Fr & Y in Fr & Same_frac(X,Y) & W in Fr & Z in Fr & Same_frac(W,Z)) imp     Same_frac([car(X) S_TIMES  car(W),cdr(X) S_TIMES  cdr(W)],[car(Y) S_TIMES  car(Z),cdr(Y) S_TIMES  cdr(Z)])"], ["TU55", "(M in Re & M /= R_0) imp (Recip(M) in Re & M R_TIMES Recip(M) = R_1)"], ["T95", "(#S in #T & #T in #R) imp (#S in #R)"], ["T195", "(T in S & Card(S) & (not Finite(S))) imp (S TIMES T = S)"], ["T295", "(X in Ra) imp ({y: y in Ra | X Ra_GT y} in Re)"], ["TU95", "     is_analytic_CF(f) imp (EXISTS ep in R | (ep  incs  R_0 & ep /= R_0 &          (FORALL crv1| (FORALL crv2|  ((is_CD_curv(crv1,R_0,R_1) & is_CD_curv(crv1,R_0,R_1)             & crv1~[R_0] = crv1~[R_1] & crv2~[R_0] = crv2~[R_1] &              (FORALL x in Interval(R_0,R_1) | (ep  incs  C_abs(crv1~[x] C_MINUS crv2~[x])))                 imp  Line_Int(f,crv1,R_0,R_1) = Line_Int(f,crv2,R_0,R_1)))))))"], ["Tproduct_order.5", "(FORALL x in o1 PROD o2 | (FORALL y in o1 PROD o2 | (FORALL z in o1 PROD o2 |                                         ((Ord1p2(x,y) & Ord1p2(y,z)) imp Ord1p2(x,z)))))"], ["T25", "S - T = {x in S | x notin T}"], ["T125", "(X /= Z) imp ({[X,Y],[Z,W]}~[X] = Y)"], ["T225", "(N in Si & M in Si) imp (S_Rev(N S_PLUS M) = S_Rev(N) S_PLUS S_Rev(M))"], ["TU25", "     (FORALL f in OM | (FORALL g | ((Finite(f) & Finite(g) & Svm(f) & one_1_map(g) & domain(f) = domain(g)) imp             sigma(f) = sigma({[y,sigma(f INV_IM {y})]: y in range(g)}))))"], ["T65", "(one_1_map(F) & one_1_map(G)) imp one_1_map(F @ G)"], ["T165", "(M /= 0) imp (N OVER M incin N)"], ["T265", "(X in Ra & Y in Si & Z in Si & Z /= [0,0]) imp (Fr_to_Ra([Y,Z]) Ra_PLUS X =     Fr_to_Ra([(car(arb(X)) S_TIMES  Z) S_PLUS (cdr(arb(X)) S_TIMES  Y),(cdr(arb(X)) S_TIMES  Z)]))"], ["TU65", "(X in Re & Y in Re & Y R_TIMES Y = X & R_is_nonneg(Y)) imp (Y = sqrt(X))"], ["T35", "F @ (G @ H) = (F @ G) @ H"], ["T135", "N PROD 0 = 0 & 0 PROD N = 0"], ["T235", "(FORALL n in Si | S_Rev(n) = [0,1] S_TIMES  n)"], ["TU35", "(N in Re & M in Re) imp (N R_TIMES_ABS M = M R_TIMES_ABS N)"], ["T75", "         (ord(S) & T incin S & X in S) imp (enum(X,T)  incs  X)"], ["T175", "(N in Z & K in N & M  incs  N) imp (M MINUS N in M MINUS K)"], ["T275", "(K in Ra & N in Ra & M in Ra) imp (N Ra_TIMES (M Ra_PLUS K) = (N Ra_TIMES M) Ra_PLUS (N Ra_TIMES K))"], ["TU75", "(N in Cm) imp (C_abs(N) in Re & R_is_nonneg(C_abs(N)))"], ["Tfcn_symbol.5", "range(g) = {f(x) : x in s}"], ["T105", "Finite(S) eq Finite(#S)"], ["T205", "(K in Si & N in Z & M in Z) imp (K S_PLUS [N,M] = [N,M] S_PLUS K)"], ["T5", "car([X,Y]) = X"], ["T45", "(Svm(F) & X in F) imp (F~[car(X)] = cdr(X))"], ["T145", "(A * B = 0) imp ((X PROD A) * (Y PROD B) = 0)"], ["T245", "(FORALL x in Fr | (FORALL y in Fr | (Same_frac(x,y) eq Same_frac(y,x)) & Same_frac(x,x)))"], ["TU45", "(N in Re & M in Re) imp (abs(N) R_TIMES abs(M) = abs(N R_TIMES M))"], ["T85", "Svm(F) imp (#range(F) incin #domain(F))"], ["T185", "(N in Z & M in Z & K in Z) imp ((N  incs  M) eq ((N PLUS K)  incs  (M PLUS K)))"], ["TU85", "(M in Cm) imp (M = M C_TIMES C_1)"], ["Twell_ordered_set.5", "   (Ord(U) & Ord(V) & orden(U) /= s & orden(V) /= s) imp (Ordrel(orden(U),orden(V)) eq U in V)"], ["T19", "(enum(X,S) = S & Y  incs  X) imp (enum(Y,S) = S)"], ["T119", "Ord(0) & 0 in Z & 1 in Z & 2 in Z & 3 in Z"], ["T219", "     (N in Si) imp (S_Rev(N) in Si & S_Rev(N) S_PLUS N = [0,0] & S_Rev(S_Rev(N)) = N)"], ["TU19", "sigma(0) = e"], ["T59", "Svm(F) imp (F @ inv(F) = ident(range(F)))"], ["TU9", "rng incin s"], ["T159", "(N * M = 0) imp (N PLUS M = #(N + M))"], ["T259", "(X in Fr & Y in Fr & Same_frac(X,Y) & is_nonneg(cdr(X)) & is_nonneg(cdr(Y))) imp                         ((is_nonneg(car(X)) or car(X) = [0,0]) eq (is_nonneg(car(Y)) or car(Y) = [0,0]))"], ["TU59", "(X in Re & Y in Re & R_is_nonneg(X) & R_is_nonneg(Y) & X R_PLUS Y = R_0) imp (X = R_0 & Y = R_0)"], ["T99", "Svm(F) imp (one_1_map(F) eq (FORALL x in domain(F) | (FORALL y in domain(F) | ((F~[x] = F~[y]) imp (x = y)))))"], ["T199", "(N in Si & M in Si) imp (N S_PLUS M in Si & N S_TIMES  M in Si)"], ["T299", "(N in Re & M in Re) imp (N R_PLUS M in Re)"], ["T29", "(G incin F & one_1_map(F)) imp one_1_map(G)"], ["T129", "N TIMES M = #N TIMES #M"], ["T229", "(K in Si) imp (K S_MINUS K = [0,0])"], ["TU29", "(N in Si & M in Si & M /= [0,0] & is_nonneg(M)) imp         (EXISTS k in Si | is_nonneg(N S_MINUS (k S_TIMES  M)) & is_nonneg(((k S_PLUS [1,0])S_TIMES  M)) S_MINUS N)"], ["T69", "(F = {[[x,y],[y,x]]: x in A, y in B}) imp             (one_1_map(F) & domain(F) = (A PROD B) & range(F) = (B PROD A))"], ["T169", "(M in Z & N in Z & N /= 0) imp (M in M PLUS N)"], ["T269", "(K in Ra & N in Ra & M in Ra) imp (N Ra_PLUS (M Ra_PLUS K) = (N Ra_PLUS M) Ra_PLUS K)"], ["TU69", "(N in Cm & M in Cm) imp (N C_PLUS M = M C_PLUS N)"], ["T39", "Svm(F) eq (F = {[x,F~[x]]: x in domain(F)})"], ["T139", "#({C} PROD N) = #N"], ["T239", "(Ord(S)) imp (Ult_membs(S) = S)"], ["TU39", "(N in Re) imp (abs(N) = abs(R_Rev(N)))"], ["T79", "     Svm(F) imp (EXISTS h | (domain(h) = range(F) & range(h) incin domain(F) & one_1_map(h)) &                             (FORALL x in range(F) | (F~[h~[x]] = x)))"], ["T179", "#{S} = {{}}"], ["T279", "(N in Ra & M in Ra & M /= Ra_0) imp (N = M Ra_TIMES (N Ra_OVER M))"], ["TU79", "(N in Cm & M in Cm) imp (C_abs(N) C_TIMES C_abs(M) = C_abs(N C_TIMES M))"], ["T109", "                     (Ord(S) & Ord(T) & (not Finite(S)) & Finite(T)) imp (T in S)"], ["T209", "(N in Si & M in Si) imp (N S_TIMES  M = M S_TIMES  N)"], ["T9", "(Ord(S) & Ord(T)) imp Ord(S * T)"], ["T49", "one_1_map(F) imp one_1_map(F ON  S)"], ["T149", "N PLUS (M PLUS K) = (N PLUS M) PLUS K"], ["T249", "((X in Si & Y in Si & is_nonneg(X) & is_nonneg(Y))) imp (is_nonneg(X S_PLUS Y) & is_nonneg(X S_TIMES  Y))"], ["TU49", "(K in Re & n in Re & m in Re) imp (n R_PLUS (m R_PLUS K) = (n R_PLUS m) R_PLUS K)"], ["T89", "Svm(F) imp (#domain(F) = #F)"], ["T189", "(not Finite(S)) imp (#S = #(S + {C}))"], ["TU89", "(N in RF & m in RF) imp (N F_PLUS m = m F_PLUS N)"], ["Twell_ordered_set.9", "((Ord(V) & orden(V) /= s) imp one_1_map({[x,orden(x)]: x in V})) &                 domain({[x,orden(x)]: x in V}) = V & range({[x,orden(x)]: x in V}) = {u in s: Ordrel(u,orden(V))} &         {u in s: Ordrel(u,orden(V))} = {orden(x): x in V}"], ["T12", "(Ord(S) & T in S) imp Ord(T)"], ["T112", "(one_1_map(F) & X in domain(F) & Y in domain(F) & F~[X] = F~[Y]) imp (X = Y)"], ["T212", "(N in Z) imp (Red([N,0]) = [N,0])"], ["TU12", "m in Z & P(m) & (FORALL n in m | not P(n))"], ["T52", "     (one_1_map(F) & one_1_map(G) & range(F) * range(G) = 0 & domain(F) * domain(G) = 0) imp one_1_map(F + G)"], ["TU2", "(s /= 0) imp ({e(x): x  in s} /= 0)"], ["T152", "(Finite(N) & Finite(M)) imp Finite(N TIMES M)"], ["T252", "X in Fr eq (X = [car(X),cdr(X)] & car(X) in Si & cdr(X) in Si & cdr(X) /= [0,0])"], ["TU52", "(K in Re & n in Re & m in Re) imp (n R_TIMES (m R_PLUS K) = (n R_TIMES m) R_PLUS (n R_TIMES K))"], ["T92", "Card(S) eq (S = #S)"], ["T192", "(not Finite(S)) imp (S PLUS T = #S + #T & #(S + T) = #S + #T)"], ["T292", "Ra_1 Ra_GT Ra_0"], ["TU92", "(K in RF & n in RF & m in RF) imp (n F_PLUS (m F_PLUS K) = (n F_PLUS m) F_PLUS K)"], ["Tproduct_order.2", "(FORALL x in o1 PROD o2 | Ord(cdr(x)))"], ["T22", "     (FORALL s | (EXISTS x | (Ord(x) & s = {enum(y,s): y in x}) &                             (FORALL y in x | (FORALL z in x | ((y /= z) imp (enum(y,s) /= enum(z,s)))))))"], ["T122", "{[x,0]: x in N} * {[x,1]: x in M} = 0"], ["T222", "(K in Si & N in Si & M in Si) imp (N S_TIMES  (M S_TIMES  K) = (N S_TIMES  M) S_TIMES  K)"], ["TU22", "(Finite(F) & range(F) incin s & C in F) imp (sigma(F) = sigma(F - {C}) PLUZ cdr(C))"], ["Ttransfinite_induction.2", "P(m) & (FORALL k in m | not P(k))"], ["T62", "is_map(F @ G)"], ["T162", "(#M in #N or #M = #N) imp (#N = #M PLUS (#N MINUS #M))"], ["T262", "(X in Fr) imp (Ra_is_nonneg(X) eq Ra_is_nonneg([S_Rev(car(X)),S_Rev(cdr(X))]))"], ["TU62", "(X in Re & Y in Re & X R_GT Y) imp         (X R_GT (X R_PLUS Y) R_OVER (R_1 + R_1) & (X R_PLUS Y) R_OVER (R_1 + R_1) R_GT Y)"], ["T32", "A * B = {x in A | x in B}"], ["T132", "Finite(N) & Finite(M) eq Finite(N + M)"], ["T232", "             (FORALL n in Si | (FORALL m in Si | (m S_TIMES  n = [0,0]) imp (m = [0,0] or n = [0,0])))"], ["TU32", "(N in Re & M in Re) imp (N + M in Re)"], ["T72", "     (EXISTS o | Ord(o) & S = {enum(x,S): x in o} & #o = #S)"], ["T172", "(M incin N) imp (M PLUS K incin N PLUS K)"], ["T272", "(N in Ra & M in Ra) imp (N = M Ra_PLUS (N Ra_MINUS M))"], ["TU72", "(N in Cm) imp (N C_PLUS C_Rev(N) = C_0)"], ["Tfcn_symbol.2", "(FORALL x in s | g~[x] = f(x))"], ["T102", "one_1_map(F) imp ((Finite(domain(F))) imp (Finite(range(F))))"], ["T202", "(J in Z & K in Z & N in Z & M in Z) imp ([J,K] S_PLUS [N,M] = [J,K] S_PLUS Red([N,M]))"], ["T302", "(N in Re) imp (R_Rev(N) in Re)"], ["T2", "arb([X,Y]) = {X}"], ["T42", "domain(F + G) = domain(F) + domain(G)"], ["T142", "N TIMES 1 = #N"], ["T242", "(Y in Ult_membs(S)) imp (Ult_membs(Y) incin Ult_membs(S))"], ["TU42", "(N in Re & M in Re) imp (N R_PLUS abs(M) R_GT n or n R_PLUS abs(M) = n)"], ["T82", "             (Card(C)  & (EXISTS f in OM | (one_1_map(f) & range(f) = S & domain(f) = C))) imp (C = #S)"], ["T182", "(N  incs  M) imp ((N MINUS K)  incs  (M MINUS K))"], ["T282", "(X in Ra) imp (X = X Ra_TIMES Ra_1)"], ["TU82", "(K in Cm & n in Cm & m in Cm) imp (n C_PLUS (m C_PLUS K) = (n C_PLUS m) C_PLUS K)"], ["Twell_ordered_set.2", "s incin {orden(y): y in X} eq orden(X) = s"], ["T16", "(Ord(S) & Ord(T)) imp (((T incin S) eq (T in S)) or T = S)"], ["T116", "not Finite(#s_inf)"], ["T216", "(N in Si & M in Si) imp                 (S_Rev(N S_TIMES  M) = S_Rev(N) S_TIMES  M & S_Rev(N S_TIMES  M)  = N S_TIMES  S_Rev(M))"], ["TU16", "     (EXISTS h | (FORALL n in Z | (FORALL s | (FORALL x | ((#x incin n) imp (h(x,s) = f({g4(h(y,s),x,y,s): y incin x | y /= x & P(x,y,s)},x,s)))))))"], ["T56", "one_1_map(F) imp (FORALL x in domain(F) | (inv(F)~[F~[x]] = x))"], ["TU6", "#{[x,f(x)]: x in s} = #s & #{f(x): x in s} incin #s"], ["T156", "#N in #pow(N)"], ["T256", "(X in Ra & Y in Si & Z in Si & Z /= [0,0]) imp (X Ra_PLUS Fr_to_Ra([Y,Z]) =     Fr_to_Ra([(car(arb(X)) S_TIMES  Z) S_PLUS (cdr(arb(X)) S_TIMES  Y),(cdr(arb(X)) S_TIMES  Z)]))"], ["TU56", "(N in Re & M in Re & M /= R_0) imp (N = M R_TIMES (N R_OVER M))"], ["T96", "#((A PROD B) PROD C) = #(A PROD (B PROD C))"], ["T196", "(M in Z & N in Z) imp (Red([M,N]) in Si & M * N in Z)"], ["T296", "R_0 in Re & R_1 in Re & R_is_nonneg(R_0) & R_is_nonneg(R_1) & R_1 R_GT R_0"], ["TU96", "     (is_analytic_CF(f) & domain(f) incs {z in C: C_abs(z) incin R_1}) imp         ((FORALL z in C | (((C_abs(z) incin R_1) and (C_abs(z) /= R_1)) imp  f~[z] = Line_Int({[x,f~[x] C_OVER (x C_MINUS z)]: x in C -{z}},                     {[x,C_exp_fcn([R_0,x])]: x in R},R_0,pi R_PLUS pi)                                                     C_OVER ([R_0,pi R_PLUS pi]))))"], ["Tproduct_order.6", "(T incin o1 PROD o2 & T /= 0) imp (EXISTS x in T | (FORALL y in t | (Ord1p2(x,y) or (x = y))))"], ["T26", "is_map(F) eq (FORALL x in F | x = [car(x),cdr(x)])"], ["T126", "#{[x,0]: x in M} = #M & #{[x,1]: x in N} = #N"], ["T226", "S_TIMES  [0,1] = [1,0]"], ["TU26", "(FORALL x in s | (FORALL y in s | P(x,y) eq f(x) = f(y)))"], ["T66", "(F + FF) @ G = (F @ G) + (FF @ G)"], ["T166", "(M /= 0 & N in Z) imp (N OVER M in Z & N OVER M incin N)"], ["T266", "(X in Ra & Y in Si & Z in Si & Z /= [0,0]) imp (Fr_to_Ra([Y,Z]) Ra_PLUS X =     Fr_to_Ra([(car(arb(X)) S_TIMES  Z) S_PLUS (cdr(arb(X)) S_TIMES  Y),(cdr(arb(X)) S_TIMES  Z)]))"], ["TU66", "(X in Re & R_is_nonneg(X) & Y in Re & R_is_nonneg(Y)) imp (sqrt(X R_TIMES Y) = sqrt(X) R_TIMES sqrt(Y))"], ["T36", "(F + G) ON  A = (F ON  A) + (G ON  A)"], ["T136", "N TIMES 0 = 0"], ["T236", "S incin Ult_membs(S)"], ["TU36", "(N in Re & M in Re) imp (N R_TIMES M = M R_TIMES N)"], ["T76", "(Ord(S) & T incin S) imp ({enum(x,T): x in S} incs  T)"], ["T176", "(M in Z & N in Z & K in Z & N  incs  M & N MINUS M  incs  K) imp                             (N  incs  M PLUS K & N MINUS (M PLUS K) = (N MINUS M) MINUS K)"], ["T276", "(X in Si & Y in Si & Y /= [0,0]) imp (Ra_is_nonneg(Fr_to_Ra([X,Y])) eq is_nonneg(X S_TIMES  Y))"], ["TU76", "(N in Cm) imp (C_abs(N) = C_abs(C_Rev(N)))"], ["Tfcn_symbol.6", "(FORALL x in s | (FORALL y in s | ((f(x) = f(y)) imp (x = y)))) imp one_1_map(g)"], ["T106", "(Finite(S) & T incin S & T /= S) imp (#T in #S)"], ["T206", "(J in Z & K in Z & N in Z & M in Z) imp ([J,K] S_PLUS [N,M] = [N,M] S_PLUS [J,K])"], ["T6", "cdr([X,Y]) = Y"], ["T46", "     (Svm(F) & Svm(G) & domain(F) * domain(G) = 0) imp Svm(F + G)"], ["T146", "N PLUS M = M PLUS N"], ["T246", "(FORALL x in Fr | (FORALL y in Fr | (FORALL z in Fr |                     ((Same_frac(x,y) & Same_frac(y,z)) imp Same_frac(x,z)))))"], ["TU46", "(N in Re & M in Re & M /= R_0) imp (abs(N) R_OVER abs(M) = abs(N R_OVER M))"], ["T86", "Ord(0) & Finite(0) & Card(0)"], ["T186", "(N  incs  M) imp (#N = #M PLUS #(N - M))"], ["TU86", "(M in Cm & M /= C_0) imp (C_Recip(M) in Cm & M C_TIMES C_Recip(M) = C_1)"], ["Twell_ordered_set.6", "(Ord(V) & orden(V) /= s) imp ({u in s: Ordrel(u,orden(V))} = {orden(x): x in V})"], ["T13", "not(EXISTS x | (FORALL y | y in x))"], ["T113", "Finite(S) eq Finite(S + {X})"], ["T213", "       (N in Z & M in Z) imp ([N PLUS M,0] = [N,0] S_PLUS [M,0] & [N TIMES M,0] = [N,0] S_TIMES  [M,0] &                                                         (N  incs  M) imp ([N,0] S_MINUS [M,0] = [N MINUS M,0]))"], ["TU13", "R(m,j) & (FORALL k in m | (FORALL j in o | not R(k,j)) & (FORALL i in j | not R(m,i)))"], ["T53", "is_map(inv(F)) & range(inv(F)) = domain(F) & domain(inv(F)) = range(F)"], ["TU3", "({x in s | P(x)} /= 0) imp ({e(x): x  in s | P(x)} /= 0)"], ["T153", "((Finite(N) & Finite(M)) or N = 0 or M = 0) eq Finite(N TIMES M)"], ["T253", "(N in Ra) imp (arb(N) in Fr & arb(N) = [car(arb(N)),cdr(arb(N))] & car(arb(N)) in Si &                                                             cdr(arb(N)) in Si & cdr(arb(N)) /= [0,0])"], ["TU53", "X in Re & Y in Re & R_is_nonneg(x) & R_is_nonneg(y)                 imp (R_is_nonneg(x R_PLUS y) & R_is_nonneg(x R_TIMES y))"], ["T93", "#S = ##S"], ["T193", "(not Finite(S)) imp (EXISTS T | #(T PROD T) = #S)"], ["T293", "(X in Ra & X Ra_GT Ra_0) imp (Recip(X) Ra_GT Ra_0)"], ["TU93", "(K in RF & n in RF & m in RF) imp (n F_TIMES (m F_TIMES K) = (n F_TIMES m) F_TIMES K)"], ["Tproduct_order.3", "(FORALL x in o1 PROD o2 | Ord(car(x) + cdr(x)))"], ["T23", "F ON  A incin F"], ["T123", "is_map(0) & Svm(0) & one_1_map(0) & range(0) = 0 & domain(0) = 0"], ["T223", "(N in Si & M in Si) imp (N S_MINUS M = N S_PLUS S_Rev(M))"], ["TU23", "(Finite(F) & is_map(F) & range(F) incin s) imp         (FORALL t | (sigma(F) = sigma(F ON  (domain(F) * t)) PLUZ sigma(F ON  (domain(F) - t))))"], ["T63", "(Svm(F) & Svm(G)) imp Svm(F @ G)"], ["T163", "(FORALL x in S | x incin Un(S)) & ((FORALL x in S | x incin T) imp (Un(S) incin T))"], ["T263", "(X in Fr & Y in Fr & Same_frac(X,Y)) imp (Ra_is_nonneg(X) eq Ra_is_nonneg(Y))"], ["TU63", "(S /= 0 & S incin Re) imp (Un(S) in Re or Un(S) = Ra)"], ["T33", "(is_map(F) & is_map(G)) imp is_map(F + G)"], ["T133", "Finite(N PLUS M) eq Finite(N + M)"], ["T233", "     (FORALL n in Si | (FORALL m in Si | (FORALL k in Si | (m S_TIMES  n) S_MINUS (k S_TIMES  n) = (m S_MINUS k) S_TIMES  n)))"], ["TU33", "(N in Re) imp (abs(N) in Re & N incin abs(N))"], ["T73", "     (Ord(R) & R  incs  S & S  incs  T) imp (arb(S) in arb(T) or arb(S) = arb(T) or T = 0)"], ["T173", "(M incin N) imp (M TIMES K incin N TIMES K)"], ["T273", "(K in Ra & N in Ra & M in Ra) imp (N Ra_TIMES (M Ra_TIMES K) = (N Ra_TIMES M) Ra_TIMES K)"], ["TU73", "(N in Cm & M in Cm) imp (N = M C_PLUS (N C_MINUS M))"], ["Tfcn_symbol.3", "(X notin s) imp (g~[X] = 0)"], ["T103", "one_1_map(F) imp (Finite(domain(F)) eq Finite(range(F)))"], ["T203", "(K in Si & N in Z & M in Z) imp (K S_PLUS [N,M] = K S_PLUS Red([N,M]))"], ["T3", "arb(arb([X,Y])) = X"], ["T43", "range(F + G) = range(F) + range(G)"], ["T143", "(M /= 0) imp (#(N PROD M)  incs  #N)"], ["T243", "(Y in Ult_membs(S)) imp (Y incin Ult_membs(S))"], ["TU43", "(N in Re & M in Re) imp (abs(N) R_PLUS abs(M) R_GT abs(N R_PLUS M) or abs(N) R_PLUS abs(M) = abs(N R_PLUS M))"], ["T83", "(T incin S) imp (#T incin #S)"], ["T183", "(Finite(N) & N  incs  M) imp (#(N - M) = #(#N - #M))"], ["T283", "(X in Ra) imp (X = Ra_0 eq car(arb(X)) = [0,0])"], ["TU83", "(K in Cm & n in Cm & m in Cm) imp (n C_TIMES (m C_TIMES K) = (n C_TIMES m) C_TIMES K)"], ["Twell_ordered_set.3", "(orden(X) /= s) imp (orden(X) in s)"], ["T17", "(Ord(X) & S in {enum(y,S): y in X}) imp (S incin {enum(y,S): y in X})"], ["T117", "(Ord(X) & Finite(X)) imp Card(X)"], ["T217", "(K in Si) imp (S_Rev(S_Rev(K)) = K)"], ["TU17", "Finite(X) imp (h(X,S) = f({g4(h(y,S),X,y,S): y incin X | y /= X & P(X,y,S)},X,S))"], ["T57", "one_1_map(F) imp             ((FORALL x in domain(F) | (inv(F)~[F~[x]] = x)) & (FORALL x in range(F) | (F~[inv(F)~[x]] = x)))"], ["TU7", "(EXISTS m | m incin n & P(m) & (FORALL k incin m | ((k /= m) imp (not P(k)))))"], ["T157", "N MINUS N = 0"], ["T257", "(X in Ra & Y in Si & Z in Si & Z /= [0,0]) imp (X Ra_TIMES Fr_to_Ra([Y,Z]) =                                             Fr_to_Ra([car(arb(X)) S_TIMES  Y,cdr(arb(X)) S_TIMES  Z]))"], ["TU57", "   (X in Re) imp ((R_is_nonneg(X) or R_is_nonneg(R_Rev(X))) & ((R_is_nonneg(X) & R_is_nonneg(R_Rev(X))) imp (X = R_0)))"], ["T97", "#(A PROD B) = #(B PROD A)"], ["T197", "(N in Si) imp (N = [car(N),cdr(N)] & (car(N) = 0 or cdr(N) = 0) & car(N) in Z & cdr(N) in Z &                                             Red(N) = N & car(N) * cdr(N) in Z)"], ["T297", "(N in Re) imp (N incin Ra)"], ["T27", "(G incin F & is_map(F)) imp is_map(G)"], ["T127", "N PLUS M = #N PLUS #M"], ["T227", "(K in Si) imp (K S_TIMES  [1,0] = K)"], ["TU27", "(FORALL x in s | f(x) in Eqc) & (FORALL y in Eqc | g1(y) in s & f(arb(y))  = y)"], ["T67", "G @ (F + FF) = (G @ F) + (G @ FF)"], ["T167", "(N in Z & M in Z) imp (N PLUS M in Z & N TIMES M in Z & N MINUS M in Z)"], ["T267", "(N in Ra & M in Ra) imp (N Ra_TIMES M = M Ra_TIMES N)"], ["TU67", "((X in Re & Y in Re) imp ([X,Y] in Cm)) & ((m in Cm) imp (m = [car(m),cdr(m)] & car(m) in Re & cdr(m) in Re))"], ["T37", "F ON  domain(F) = F"], ["T137", "0 TIMES N = 0"], ["T237", "Ult_membs(S) = S + {y: x in S, y in Ult_membs(x)}"], ["TU37", "(N in Re) imp (abs(N) = if R_is_nonneg(N) then N else R_Rev(N) end if)"], ["T77", "(Ord(S) & T incin S) imp (EXISTS x incin S | (Ord(x) & T = {enum(y,T): y in x} &                             (FORALL y in x | (FORALL z in x | ((y /= z) imp (enum(y,T) /= enum(z,T)))))))"], ["T177", "(M in Z & N in Z) imp ((M PLUS N) MINUS N = M)"], ["T277", "(M in Ra) imp (M = M Ra_TIMES Ra_1)"], ["TU77", "(N in Cm & M in Cm) imp         ((C_abs(N) C_PLUS C_abs(M)) R_GT C_abs(N C_PLUS M) or (C_abs(N) C_PLUS C_abs(M) = C_abs(N C_PLUS M)))"], ["T107", "Finite(S) eq (not(EXISTS f in OM | Svm(f) & range(f) = S & domain(f) incin S & S /= domain(f)))"], ["T207", "(N in Si & M in Si) imp (N S_PLUS M = M S_PLUS N)"], ["T7", "[X,Y] = [car([X,Y]),cdr([X,Y])]"], ["T47", "is_map(F) imp is_map(F ON  S)"], ["T147", "N TIMES M = M TIMES N"], ["T247", "(FORALL y in Ra | arb(y) in Fr & Fr_to_Ra(arb(y)) = y) & (FORALL x in Fr | Fr_to_Ra(x) in Ra) &                             (FORALL x in Fr | (FORALL y in Fr | Same_frac(x,y) eq Fr_to_Ra(x) = Fr_to_Ra(y))) &                                 (FORALL x in Fr | Same_frac(x,arb(Fr_to_Ra(x))))"], ["TU47", "(N in Re & M in Re) imp (N R_TIMES_ABS M in Re)"], ["T87", "#domain(F) incin #F"], ["T187", "(N in Z & M in Z & K in Z & N  incs  M) imp ((N PLUS K) MINUS (M PLUS K) = N MINUS M)"], ["TU87", "(N in Cm & M in Cm & M /= C_0) imp (N = M C_TIMES (N C_OVER M))"], ["Twell_ordered_set.7", "                         (Ord(U) & Ord(V) & orden(U) /= s & orden(V) /= s & U /= V) imp (orden(U) /= orden(V))"]}
