%   ORIGINAL: 'h4/thm/nets/SEQ_TENDS_'
% 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: !d x x0. 'h4/const/nets/tends' x x0 ('h4/const/pair/,' ('h4/const/topology/mtop' d) 'h4/const/arithmetic/>=') <=> (!e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e ==> (?N. !n. 'h4/const/arithmetic/>=' n N ==> 'h4/const/realax/real_lt' ('h4/const/topology/dist' d ('h4/const/pair/,' (x n) x0)) e))
%   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: !d x x0. 'h4/const/nets/tends' x x0 ('h4/const/pair/,' ('h4/const/topology/mtop' d) 'h4/const/arithmetic/>=') <=> (!e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e ==> (?N. !n. happ (happ 'h4/const/arithmetic/>=' n) N ==> 'h4/const/realax/real_lt' ('h4/const/topology/dist' d ('h4/const/pair/,' (happ x n) x0)) e))
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_3f63861,V_3f63857]: ![F, G]: (![X]: s(V_3f63857,happ(s(fun(V_3f63861,V_3f63857),F),s(V_3f63861,X))) = s(V_3f63857,happ(s(fun(V_3f63861,V_3f63857),G),s(V_3f63861,X))) => s(fun(V_3f63861,V_3f63857),F) = s(fun(V_3f63861,V_3f63857),G))).
fof('h4/thm/nets/SEQ_TENDS_', conjecture, ![A]: ![D, X, X0]: (p(s(bool,'h4/const/nets/tends'(s(fun('h4/type/num/num',A),X),s(A,X0),s('h4/type/pair/prod'('h4/type/topology/topology'(A),fun('h4/type/num/num',fun('h4/type/num/num',bool))),'h4/const/pair/,'(s('h4/type/topology/topology'(A),'h4/const/topology/mtop'(s('h4/type/topology/metric'(A),D))),s(fun('h4/type/num/num',fun('h4/type/num/num',bool)),'h4/const/arithmetic/>=')))))) <=> ![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)))) => ?[N]: ![N0]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',bool)),'h4/const/arithmetic/>='),s('h4/type/num/num',N0))),s('h4/type/num/num',N)))) => p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real','h4/const/topology/dist'(s('h4/type/topology/metric'(A),D),s('h4/type/pair/prod'(A,A),'h4/const/pair/,'(s(A,happ(s(fun('h4/type/num/num',A),X),s('h4/type/num/num',N0))),s(A,X0))))),s('h4/type/realax/real',E)))))))).
