%   ORIGINAL: 'h4/thm/topology/OPEN_UNOPEN_'
% 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: !'S\'' top. 'h4/const/topology/open' top 'S\'' <=> 'h4/const/pred_set/BIGUNION' ('h4/const/pred_set/GSPEC' (\P. 'h4/const/pair/,' P ('h4/const/topology/open' top P /\ 'h4/const/pred_set/SUBSET' P 'S\''))) = 'S\''
%   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: !_0. (!top 'S\'' P. ?v. (v <=> 'h4/const/topology/open' top P /\ 'h4/const/pred_set/SUBSET' P 'S\'') /\ happ (happ (happ _0 top) 'S\'') P = 'h4/const/pair/,' P v) ==> (!'S\'' top. 'h4/const/topology/open' top 'S\'' <=> 'h4/const/pred_set/BIGUNION' ('h4/const/pred_set/GSPEC' (happ (happ _0 top) 'S\'')) = 'S\'')
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_3f37579,V_3f37575]: ![F, G]: (![X]: s(V_3f37575,happ(s(fun(V_3f37579,V_3f37575),F),s(V_3f37579,X))) = s(V_3f37575,happ(s(fun(V_3f37579,V_3f37575),G),s(V_3f37579,X))) => s(fun(V_3f37579,V_3f37575),F) = s(fun(V_3f37579,V_3f37575),G))).
fof('h4/thm/topology/OPEN_UNOPEN_', conjecture, ![A]: ![V__0]: (![Top, V_27S_5c_27_27, P]: ?[V]: ((p(s(bool,V)) <=> (p(s(bool,'h4/const/topology/open'(s('h4/type/topology/topology'(A),Top),s(fun(A,bool),P)))) & p(s(bool,'h4/const/pred_set/SUBSET'(s(fun(A,bool),P),s(fun(A,bool),V_27S_5c_27_27)))))) & s('h4/type/pair/prod'(fun(A,bool),bool),happ(s(fun(fun(A,bool),'h4/type/pair/prod'(fun(A,bool),bool)),happ(s(fun(fun(A,bool),fun(fun(A,bool),'h4/type/pair/prod'(fun(A,bool),bool))),happ(s(fun('h4/type/topology/topology'(A),fun(fun(A,bool),fun(fun(A,bool),'h4/type/pair/prod'(fun(A,bool),bool)))),V__0),s('h4/type/topology/topology'(A),Top))),s(fun(A,bool),V_27S_5c_27_27))),s(fun(A,bool),P))) = s('h4/type/pair/prod'(fun(A,bool),bool),'h4/const/pair/,'(s(fun(A,bool),P),s(bool,V)))) => ![V_27S_5c_27_27, Top]: (p(s(bool,'h4/const/topology/open'(s('h4/type/topology/topology'(A),Top),s(fun(A,bool),V_27S_5c_27_27)))) <=> s(fun(A,bool),'h4/const/pred_set/BIGUNION'(s(fun(fun(A,bool),bool),'h4/const/pred_set/GSPEC'(s(fun(fun(A,bool),'h4/type/pair/prod'(fun(A,bool),bool)),happ(s(fun(fun(A,bool),fun(fun(A,bool),'h4/type/pair/prod'(fun(A,bool),bool))),happ(s(fun('h4/type/topology/topology'(A),fun(fun(A,bool),fun(fun(A,bool),'h4/type/pair/prod'(fun(A,bool),bool)))),V__0),s('h4/type/topology/topology'(A),Top))),s(fun(A,bool),V_27S_5c_27_27))))))) = s(fun(A,bool),V_27S_5c_27_27)))).
