%   ORIGINAL: 'h4/thm/topology/MTOP_OPEN_'
% 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\'' m. 'h4/const/topology/open' ('h4/const/topology/mtop' m) 'S\'' <=> (!x. 'S\'' x ==> (?e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e /\ (!y. 'h4/const/realax/real_lt' ('h4/const/topology/dist' m ('h4/const/pair/,' x y)) e ==> 'S\'' y)))
%   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\'' m. 'h4/const/topology/open' ('h4/const/topology/mtop' m) 'S\'' <=> (!x. happ 'S\'' x ==> (?e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e /\ (!y. 'h4/const/realax/real_lt' ('h4/const/topology/dist' m ('h4/const/pair/,' x y)) e ==> happ 'S\'' y)))
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_3f37880,V_3f37876]: ![F, G]: (![X]: s(V_3f37876,happ(s(fun(V_3f37880,V_3f37876),F),s(V_3f37880,X))) = s(V_3f37876,happ(s(fun(V_3f37880,V_3f37876),G),s(V_3f37880,X))) => s(fun(V_3f37880,V_3f37876),F) = s(fun(V_3f37880,V_3f37876),G))).
fof('h4/thm/topology/MTOP_OPEN_', conjecture, ![A]: ![V_27S_5c_27_27, M]: (p(s(bool,'h4/const/topology/open'(s('h4/type/topology/topology'(A),'h4/const/topology/mtop'(s('h4/type/topology/metric'(A),M))),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)))) => ?[E]: (p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))),s('h4/type/realax/real',E)))) & ![Y]: (p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real','h4/const/topology/dist'(s('h4/type/topology/metric'(A),M),s('h4/type/pair/prod'(A,A),'h4/const/pair/,'(s(A,X),s(A,Y))))),s('h4/type/realax/real',E)))) => p(s(bool,happ(s(fun(A,bool),V_27S_5c_27_27),s(A,Y))))))))).
