%   ORIGINAL: 'h4/thm/pair/PAIR_FUN_THM_'
% 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. 'h4/const/bool/?!' (\f. P f) <=> 'h4/const/bool/?!' (\p. P (\a. 'h4/const/pair/,' ('h4/const/pair/FST' p a) ('h4/const/pair/SND' p a)))
%   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: !_2. (!p a. happ (happ _2 p) a = 'h4/const/pair/,' (happ ('h4/const/pair/FST' p) a) (happ ('h4/const/pair/SND' p) a)) ==> (!_1. (!P p. happ (happ _1 P) p <=> happ P (happ _2 p)) ==> (!_0. (!P f. happ (happ _0 P) f <=> happ P f) ==> (!P. 'h4/const/bool/?!' (happ _0 P) <=> 'h4/const/bool/?!' (happ _1 P))))
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_3f61611,V_3f61607]: ![F, G]: (![X]: s(V_3f61607,happ(s(fun(V_3f61611,V_3f61607),F),s(V_3f61611,X))) = s(V_3f61607,happ(s(fun(V_3f61611,V_3f61607),G),s(V_3f61611,X))) => s(fun(V_3f61611,V_3f61607),F) = s(fun(V_3f61611,V_3f61607),G))).
fof('h4/thm/pair/PAIR_FUN_THM_', conjecture, ![A,B,C]: ![V__2]: (![P, A0]: s('h4/type/pair/prod'(B,C),happ(s(fun(A,'h4/type/pair/prod'(B,C)),happ(s(fun('h4/type/pair/prod'(fun(A,B),fun(A,C)),fun(A,'h4/type/pair/prod'(B,C))),V__2),s('h4/type/pair/prod'(fun(A,B),fun(A,C)),P))),s(A,A0))) = s('h4/type/pair/prod'(B,C),'h4/const/pair/,'(s(B,happ(s(fun(A,B),'h4/const/pair/FST'(s('h4/type/pair/prod'(fun(A,B),fun(A,C)),P))),s(A,A0))),s(C,happ(s(fun(A,C),'h4/const/pair/SND'(s('h4/type/pair/prod'(fun(A,B),fun(A,C)),P))),s(A,A0))))) => ![V__1]: (![P, P0]: s(bool,happ(s(fun('h4/type/pair/prod'(fun(A,B),fun(A,C)),bool),happ(s(fun(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),fun('h4/type/pair/prod'(fun(A,B),fun(A,C)),bool)),V__1),s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),P))),s('h4/type/pair/prod'(fun(A,B),fun(A,C)),P0))) = s(bool,happ(s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),P),s(fun(A,'h4/type/pair/prod'(B,C)),happ(s(fun('h4/type/pair/prod'(fun(A,B),fun(A,C)),fun(A,'h4/type/pair/prod'(B,C))),V__2),s('h4/type/pair/prod'(fun(A,B),fun(A,C)),P0))))) => ![V__0]: (![P, F]: s(bool,happ(s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),happ(s(fun(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),fun(fun(A,'h4/type/pair/prod'(B,C)),bool)),V__0),s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),P))),s(fun(A,'h4/type/pair/prod'(B,C)),F))) = s(bool,happ(s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),P),s(fun(A,'h4/type/pair/prod'(B,C)),F))) => ![P]: s(bool,'h4/const/bool/?!'(s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),happ(s(fun(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),fun(fun(A,'h4/type/pair/prod'(B,C)),bool)),V__0),s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),P))))) = s(bool,'h4/const/bool/?!'(s(fun('h4/type/pair/prod'(fun(A,B),fun(A,C)),bool),happ(s(fun(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),fun('h4/type/pair/prod'(fun(A,B),fun(A,C)),bool)),V__1),s(fun(fun(A,'h4/type/pair/prod'(B,C)),bool),P))))))))).
