%   ORIGINAL: 'h4/thm/ind_type/INJ_INVERSE2_'
% 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. (!x1 y1 x2 y2. P x1 y1 = P x2 y2 <=> x1 = x2 /\ y1 = y2) ==> (?X Y. !x y. X (P x y) = x /\ Y (P x y) = y)
%   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: !P. (!x1 y1 x2 y2. happ (happ P x1) y1 = happ (happ P x2) y2 <=> x1 = x2 /\ y1 = y2) ==> (?X Y. !x y. happ X (happ (happ P x) y) = x /\ happ Y (happ (happ P x) y) = y)
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_3f73789,V_3f73785]: ![F, G]: (![X]: s(V_3f73785,happ(s(fun(V_3f73789,V_3f73785),F),s(V_3f73789,X))) = s(V_3f73785,happ(s(fun(V_3f73789,V_3f73785),G),s(V_3f73789,X))) => s(fun(V_3f73789,V_3f73785),F) = s(fun(V_3f73789,V_3f73785),G))).
fof('h4/thm/ind_type/INJ_INVERSE2_', conjecture, ![C,A,B]: ![P]: (![X1, Y1, X2, Y2]: (s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),P),s(A,X1))),s(B,Y1))) = s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),P),s(A,X2))),s(B,Y2))) <=> (s(A,X1) = s(A,X2) & s(B,Y1) = s(B,Y2))) => ?[X, Y]: ![X0, Y0]: (s(A,happ(s(fun(C,A),X),s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),P),s(A,X0))),s(B,Y0))))) = s(A,X0) & s(B,happ(s(fun(C,B),Y),s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),P),s(A,X0))),s(B,Y0))))) = s(B,Y0)))).
