%   ORIGINAL: 'h4/thm/quotient/FUN_REL_EQ_REL_'
% 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: !R1 abs1 rep1. 'h4/const/quotient/QUOTIENT' R1 abs1 rep1 ==> (!R2 abs2 rep2. 'h4/const/quotient/QUOTIENT' R2 abs2 rep2 ==> (!f g. 'h4/const/quotient/===>' R1 R2 f g <=> 'h4/const/quotient/respects' ('h4/const/quotient/===>' R1 R2) f /\ 'h4/const/quotient/respects' ('h4/const/quotient/===>' R1 R2) g /\ 'h4/const/quotient/-->' rep1 abs2 f = 'h4/const/quotient/-->' rep1 abs2 g))
%   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: !R1 abs1 rep1. 'h4/const/quotient/QUOTIENT' R1 abs1 rep1 ==> (!R2 abs2 rep2. 'h4/const/quotient/QUOTIENT' R2 abs2 rep2 ==> (!f g. happ (happ ('h4/const/quotient/===>' R1 R2) f) g <=> 'h4/const/quotient/respects' ('h4/const/quotient/===>' R1 R2) f /\ 'h4/const/quotient/respects' ('h4/const/quotient/===>' R1 R2) g /\ 'h4/const/quotient/-->' rep1 abs2 f = 'h4/const/quotient/-->' rep1 abs2 g))
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_3f75015,V_3f75011]: ![F, G]: (![X]: s(V_3f75011,happ(s(fun(V_3f75015,V_3f75011),F),s(V_3f75015,X))) = s(V_3f75011,happ(s(fun(V_3f75015,V_3f75011),G),s(V_3f75015,X))) => s(fun(V_3f75015,V_3f75011),F) = s(fun(V_3f75015,V_3f75011),G))).
fof('h4/thm/quotient/FUN_REL_EQ_REL_', conjecture, ![C,D,A,B]: ![R1, Abs1, Rep1]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(A,fun(A,bool)),R1),s(fun(A,C),Abs1),s(fun(C,A),Rep1)))) => ![R2, Abs2, Rep2]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(B,fun(B,bool)),R2),s(fun(B,D),Abs2),s(fun(D,B),Rep2)))) => ![F, G]: (p(s(bool,happ(s(fun(fun(A,B),bool),happ(s(fun(fun(A,B),fun(fun(A,B),bool)),'h4/const/quotient/===>'(s(fun(A,fun(A,bool)),R1),s(fun(B,fun(B,bool)),R2))),s(fun(A,B),F))),s(fun(A,B),G)))) <=> (p(s(bool,'h4/const/quotient/respects'(s(fun(fun(A,B),fun(fun(A,B),bool)),'h4/const/quotient/===>'(s(fun(A,fun(A,bool)),R1),s(fun(B,fun(B,bool)),R2))),s(fun(A,B),F)))) & (p(s(bool,'h4/const/quotient/respects'(s(fun(fun(A,B),fun(fun(A,B),bool)),'h4/const/quotient/===>'(s(fun(A,fun(A,bool)),R1),s(fun(B,fun(B,bool)),R2))),s(fun(A,B),G)))) & s(fun(C,D),'h4/const/quotient/-->'(s(fun(C,A),Rep1),s(fun(B,D),Abs2),s(fun(A,B),F))) = s(fun(C,D),'h4/const/quotient/-->'(s(fun(C,A),Rep1),s(fun(B,D),Abs2),s(fun(A,B),G))))))))).
