%   ORIGINAL: 'h4/thm/pred_set/CROSS_EQNS_e1'
% 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: !s2 s1 a. 'h4/const/pred_set/CROSS' ('h4/const/pred_set/INSERT' a s1) s2 = 'h4/const/pred_set/UNION' ('h4/const/pred_set/IMAGE' (\y. 'h4/const/pair/,' a y) s2) ('h4/const/pred_set/CROSS' s1 s2)
%   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: !_0. (!a y. happ (happ _0 a) y = 'h4/const/pair/,' a y) ==> (!s2 s1 a. 'h4/const/pred_set/CROSS' ('h4/const/pred_set/INSERT' a s1) s2 = 'h4/const/pred_set/UNION' ('h4/const/pred_set/IMAGE' (happ _0 a) s2) ('h4/const/pred_set/CROSS' s1 s2))
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_3f24102,V_3f24098]: ![F, G]: (![X]: s(V_3f24098,happ(s(fun(V_3f24102,V_3f24098),F),s(V_3f24102,X))) = s(V_3f24098,happ(s(fun(V_3f24102,V_3f24098),G),s(V_3f24102,X))) => s(fun(V_3f24102,V_3f24098),F) = s(fun(V_3f24102,V_3f24098),G))).
fof('h4/thm/pred_set/CROSS_EQNS_e1', conjecture, ![A,B]: ![V__0]: (![A0, Y]: s('h4/type/pair/prod'(A,B),happ(s(fun(B,'h4/type/pair/prod'(A,B)),happ(s(fun(A,fun(B,'h4/type/pair/prod'(A,B))),V__0),s(A,A0))),s(B,Y))) = s('h4/type/pair/prod'(A,B),'h4/const/pair/,'(s(A,A0),s(B,Y))) => ![S2, S1, A0]: s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pred_set/CROSS'(s(fun(A,bool),'h4/const/pred_set/INSERT'(s(A,A0),s(fun(A,bool),S1))),s(fun(B,bool),S2))) = s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pred_set/UNION'(s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pred_set/IMAGE'(s(fun(B,'h4/type/pair/prod'(A,B)),happ(s(fun(A,fun(B,'h4/type/pair/prod'(A,B))),V__0),s(A,A0))),s(fun(B,bool),S2))),s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pred_set/CROSS'(s(fun(A,bool),S1),s(fun(B,bool),S2))))))).
