%   ORIGINAL: 'h4/thm/topology/BALL_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: !m x e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e ==> 'h4/const/topology/neigh' ('h4/const/topology/mtop' m) ('h4/const/pair/,' ('h4/const/topology/B' m ('h4/const/pair/,' x e)) x)
%   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: !m x e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e ==> 'h4/const/topology/neigh' ('h4/const/topology/mtop' m) ('h4/const/pair/,' ('h4/const/topology/B' m ('h4/const/pair/,' x e)) x)
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_3f37930,V_3f37926]: ![F, G]: (![X]: s(V_3f37926,happ(s(fun(V_3f37930,V_3f37926),F),s(V_3f37930,X))) = s(V_3f37926,happ(s(fun(V_3f37930,V_3f37926),G),s(V_3f37930,X))) => s(fun(V_3f37930,V_3f37926),F) = s(fun(V_3f37930,V_3f37926),G))).
fof('h4/thm/topology/BALL_NEIGH_', conjecture, ![A]: ![M, 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)))) => p(s(bool,'h4/const/topology/neigh'(s('h4/type/topology/topology'(A),'h4/const/topology/mtop'(s('h4/type/topology/metric'(A),M))),s('h4/type/pair/prod'(fun(A,bool),A),'h4/const/pair/,'(s(fun(A,bool),'h4/const/topology/B'(s('h4/type/topology/metric'(A),M),s('h4/type/pair/prod'(A,'h4/type/realax/real'),'h4/const/pair/,'(s(A,X),s('h4/type/realax/real',E))))),s(A,X)))))))).
