%   ORIGINAL: 'h4/thm/quotient/EQUALS_EQUIV_IMPLIES_'
% 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: !b2 b1 a2 a1 R. 'h4/const/quotient/EQUIV' R ==> R a1 a2 /\ R b1 b2 ==> a1 = b1 ==> R a2 b2
%   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: !b2 b1 a2 a1 R. 'h4/const/quotient/EQUIV' R ==> happ (happ R a1) a2 /\ happ (happ R b1) b2 ==> a1 = b1 ==> happ (happ R a2) b2
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_3f76165,V_3f76161]: ![F, G]: (![X]: s(V_3f76161,happ(s(fun(V_3f76165,V_3f76161),F),s(V_3f76165,X))) = s(V_3f76161,happ(s(fun(V_3f76165,V_3f76161),G),s(V_3f76165,X))) => s(fun(V_3f76165,V_3f76161),F) = s(fun(V_3f76165,V_3f76161),G))).
fof('h4/thm/quotient/EQUALS_EQUIV_IMPLIES_', conjecture, ![A]: ![B2, B1, A2, A1, R]: (p(s(bool,'h4/const/quotient/EQUIV'(s(fun(A,fun(A,bool)),R)))) => ((p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,A1))),s(A,A2)))) & p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,B1))),s(A,B2))))) => (s(A,A1) = s(A,B1) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,A2))),s(A,B2)))))))).
