%   ORIGINAL: 'h4/thm/quotient/EQUIV_REFL_SYM_TRANS_'
% 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: !R. (!x y. R x y <=> R x = R y) <=> (!x. R x x) /\ (!x y. R x y ==> R y x) /\ (!x y z. R x y /\ R y z ==> R x z)
%   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: !R. (!x y. happ (happ R x) y <=> happ R x = happ R y) <=> (!x. happ (happ R x) x) /\ (!x y. happ (happ R x) y ==> happ (happ R y) x) /\ (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
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_3f74640,V_3f74636]: ![F, G]: (![X]: s(V_3f74636,happ(s(fun(V_3f74640,V_3f74636),F),s(V_3f74640,X))) = s(V_3f74636,happ(s(fun(V_3f74640,V_3f74636),G),s(V_3f74640,X))) => s(fun(V_3f74640,V_3f74636),F) = s(fun(V_3f74640,V_3f74636),G))).
fof('h4/thm/quotient/EQUIV_REFL_SYM_TRANS_', conjecture, ![A]: ![R]: (![X, Y]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Y)))) <=> s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))) = s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,Y)))) <=> (![X]: p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,X)))) & (![X, Y]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Y)))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,Y))),s(A,X))))) & ![X, Y, Z]: ((p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Y)))) & p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,Y))),s(A,Z))))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Z))))))))).
