%   ORIGINAL: 'h4/thm/quotient/RES_EXISTS_PRS_'
% 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 abs rep. 'h4/const/quotient/QUOTIENT' R abs rep ==> (!P f. 'h4/const/bool/RES_EXISTS' P f <=> 'h4/const/bool/RES_EXISTS' ('h4/const/quotient/-->' abs 'h4/const/combin/I' P) ('h4/const/quotient/-->' abs 'h4/const/combin/I' f))
%   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 abs rep. 'h4/const/quotient/QUOTIENT' R abs rep ==> (!P f. 'h4/const/bool/RES_EXISTS' P f <=> 'h4/const/bool/RES_EXISTS' ('h4/const/quotient/-->' abs 'h4/const/combin/I' P) ('h4/const/quotient/-->' abs 'h4/const/combin/I' f))
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_3f75490,V_3f75486]: ![F, G]: (![X]: s(V_3f75486,happ(s(fun(V_3f75490,V_3f75486),F),s(V_3f75490,X))) = s(V_3f75486,happ(s(fun(V_3f75490,V_3f75486),G),s(V_3f75490,X))) => s(fun(V_3f75490,V_3f75486),F) = s(fun(V_3f75490,V_3f75486),G))).
fof('h4/thm/quotient/RES_EXISTS_PRS_', conjecture, ![A,B]: ![R, Abs, Rep]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(A,fun(A,bool)),R),s(fun(A,B),Abs),s(fun(B,A),Rep)))) => ![P, F]: s(bool,'h4/const/bool/RES_EXISTS'(s(fun(B,bool),P),s(fun(B,bool),F))) = s(bool,'h4/const/bool/RES_EXISTS'(s(fun(A,bool),'h4/const/quotient/-->'(s(fun(A,B),Abs),s(fun(bool,bool),'h4/const/combin/I'),s(fun(B,bool),P))),s(fun(A,bool),'h4/const/quotient/-->'(s(fun(A,B),Abs),s(fun(bool,bool),'h4/const/combin/I'),s(fun(B,bool),F))))))).
