%   ORIGINAL: 'h4/thm/topology/OPEN_NEIGH_'
% 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\'' <=> (!x. 'S\'' x ==> (?N. 'h4/const/topology/neigh' top ('h4/const/pair/,' N x) /\ 'h4/const/pred_set/SUBSET' N '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: !'S\'' top. 'h4/const/topology/open' top 'S\'' <=> (!x. happ 'S\'' x ==> (?N. 'h4/const/topology/neigh' top ('h4/const/pair/,' N x) /\ 'h4/const/pred_set/SUBSET' N '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_3f37630,V_3f37626]: ![F, G]: (![X]: s(V_3f37626,happ(s(fun(V_3f37630,V_3f37626),F),s(V_3f37630,X))) = s(V_3f37626,happ(s(fun(V_3f37630,V_3f37626),G),s(V_3f37630,X))) => s(fun(V_3f37630,V_3f37626),F) = s(fun(V_3f37630,V_3f37626),G))).
fof('h4/thm/topology/OPEN_NEIGH_', conjecture, ![A]: ![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)))) <=> ![X]: (p(s(bool,happ(s(fun(A,bool),V_27S_5c_27_27),s(A,X)))) => ?[N]: (p(s(bool,'h4/const/topology/neigh'(s('h4/type/topology/topology'(A),Top),s('h4/type/pair/prod'(fun(A,bool),A),'h4/const/pair/,'(s(fun(A,bool),N),s(A,X)))))) & p(s(bool,'h4/const/pred_set/SUBSET'(s(fun(A,bool),N),s(fun(A,bool),V_27S_5c_27_27)))))))).
