%   ORIGINAL: 'h4/thm/pair/LAMBDA_PROD_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !P. (\p. P p) = 'h4/const/pair/UNCURRY' (\p1 p2. P ('h4/const/pair/,' p1 p2))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_1. (!P p1 p2. happ (happ (happ _1 P) p1) p2 = happ P ('h4/const/pair/,' p1 p2)) ==> (!_0. (!P p1. happ (happ _0 P) p1 = happ (happ _1 P) p1) ==> (!P x. happ P x = 'h4/const/pair/UNCURRY' (happ _0 P) x))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f61136,V_3f61132]: ![F, G]: (![X]: s(V_3f61132,happ(s(fun(V_3f61136,V_3f61132),F),s(V_3f61136,X))) = s(V_3f61132,happ(s(fun(V_3f61136,V_3f61132),G),s(V_3f61136,X))) => s(fun(V_3f61136,V_3f61132),F) = s(fun(V_3f61136,V_3f61132),G))).
fof('h4/thm/pair/LAMBDA_PROD_', conjecture, ![C,A,B]: ![V__1]: (![P, P1, P2]: s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),happ(s(fun(fun('h4/type/pair/prod'(A,B),C),fun(A,fun(B,C))),V__1),s(fun('h4/type/pair/prod'(A,B),C),P))),s(A,P1))),s(B,P2))) = s(C,happ(s(fun('h4/type/pair/prod'(A,B),C),P),s('h4/type/pair/prod'(A,B),'h4/const/pair/,'(s(A,P1),s(B,P2))))) => ![V__0]: (![P, P1]: s(fun(B,C),happ(s(fun(A,fun(B,C)),happ(s(fun(fun('h4/type/pair/prod'(A,B),C),fun(A,fun(B,C))),V__0),s(fun('h4/type/pair/prod'(A,B),C),P))),s(A,P1))) = s(fun(B,C),happ(s(fun(A,fun(B,C)),happ(s(fun(fun('h4/type/pair/prod'(A,B),C),fun(A,fun(B,C))),V__1),s(fun('h4/type/pair/prod'(A,B),C),P))),s(A,P1))) => ![P, X]: s(C,happ(s(fun('h4/type/pair/prod'(A,B),C),P),s('h4/type/pair/prod'(A,B),X))) = s(C,'h4/const/pair/UNCURRY'(s(fun(A,fun(B,C)),happ(s(fun(fun('h4/type/pair/prod'(A,B),C),fun(A,fun(B,C))),V__0),s(fun('h4/type/pair/prod'(A,B),C),P))),s('h4/type/pair/prod'(A,B),X)))))).
