%   ORIGINAL: 'h4/thm/relation/WeakOrder_EQ_'
% 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. 'h4/const/relation/WeakOrder' R ==> (!y z. y = z <=> R y z /\ R z 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: !R. 'h4/const/relation/WeakOrder' R ==> (!y z. y = z <=> happ (happ R y) z /\ happ (happ R z) 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_3f29678,V_3f29674]: ![F, G]: (![X]: s(V_3f29674,happ(s(fun(V_3f29678,V_3f29674),F),s(V_3f29678,X))) = s(V_3f29674,happ(s(fun(V_3f29678,V_3f29674),G),s(V_3f29678,X))) => s(fun(V_3f29678,V_3f29674),F) = s(fun(V_3f29678,V_3f29674),G))).
fof('h4/thm/relation/WeakOrder_EQ_', conjecture, ![A]: ![R]: (p(s(bool,'h4/const/relation/WeakOrder'(s(fun(A,fun(A,bool)),R)))) => ![Y, Z]: (s(A,Y) = s(A,Z) <=> (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,Z))),s(A,Y)))))))).
