%   ORIGINAL: 'h4/thm/quotient_list/FILTER_RSP_'
% 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 ==> (!P1 P2 l1 l2. 'h4/const/quotient/===>' R $equals P1 P2 /\ 'h4/const/list/LIST_REL' R l1 l2 ==> 'h4/const/list/LIST_REL' R ('h4/const/list/FILTER' P1 l1) ('h4/const/list/FILTER' P2 l2))
%   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 ==> (!P1 P2 l1 l2. 'h4/const/quotient/===>' R $equals P1 P2 /\ 'h4/const/list/LIST_REL' R l1 l2 ==> 'h4/const/list/LIST_REL' R ('h4/const/list/FILTER' P1 l1) ('h4/const/list/FILTER' P2 l2))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f67786,V_3f67782]: ![F, G]: (![X]: s(V_3f67782,happ(s(fun(V_3f67786,V_3f67782),F),s(V_3f67786,X))) = s(V_3f67782,happ(s(fun(V_3f67786,V_3f67782),G),s(V_3f67786,X))) => s(fun(V_3f67786,V_3f67782),F) = s(fun(V_3f67786,V_3f67782),G))).
fof('h4/thm/quotient_list/FILTER_RSP_', conjecture, ![B,A]: ![R, Abs, Rep]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(A,fun(A,bool)),R),s(fun(A,B),Abs),s(fun(B,A),Rep)))) => ![P1, P2, L1, L2]: ((p(s(bool,'h4/const/quotient/===>'(s(fun(A,fun(A,bool)),R),s(fun(bool,fun(bool,bool)),'$equals'),s(fun(A,bool),P1),s(fun(A,bool),P2)))) & p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(A,bool)),R),s('h4/type/list/list'(A),L1),s('h4/type/list/list'(A),L2))))) => p(s(bool,'h4/const/list/LIST_REL'(s(fun(A,fun(A,bool)),R),s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),P1),s('h4/type/list/list'(A),L1))),s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),P2),s('h4/type/list/list'(A),L2))))))))).
