%   ORIGINAL: 'h4/thm/nets/LIM_TENDS2_'
% 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: !m1 m2 f x0 y0. 'h4/const/topology/limpt' ('h4/const/topology/mtop' m1) x0 'h4/const/pred_set/UNIV' ==> ('h4/const/nets/tends' f y0 ('h4/const/pair/,' ('h4/const/topology/mtop' m2) ('h4/const/nets/tendsto' ('h4/const/pair/,' m1 x0))) <=> (!e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e ==> (?d. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') d /\ (!x. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/topology/dist' m1 ('h4/const/pair/,' x x0)) /\ 'h4/const/realax/real_lt' ('h4/const/topology/dist' m1 ('h4/const/pair/,' x x0)) d ==> 'h4/const/realax/real_lt' ('h4/const/topology/dist' m2 ('h4/const/pair/,' (f x) y0)) 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: !m1 m2 f x0 y0. 'h4/const/topology/limpt' ('h4/const/topology/mtop' m1) x0 'h4/const/pred_set/UNIV' ==> ('h4/const/nets/tends' f y0 ('h4/const/pair/,' ('h4/const/topology/mtop' m2) ('h4/const/nets/tendsto' ('h4/const/pair/,' m1 x0))) <=> (!e. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') e ==> (?d. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') d /\ (!x. 'h4/const/realax/real_lt' ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/topology/dist' m1 ('h4/const/pair/,' x x0)) /\ 'h4/const/realax/real_lt' ('h4/const/topology/dist' m1 ('h4/const/pair/,' x x0)) d ==> 'h4/const/realax/real_lt' ('h4/const/topology/dist' m2 ('h4/const/pair/,' (happ f x) y0)) 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_3f63911,V_3f63907]: ![F, G]: (![X]: s(V_3f63907,happ(s(fun(V_3f63911,V_3f63907),F),s(V_3f63911,X))) = s(V_3f63907,happ(s(fun(V_3f63911,V_3f63907),G),s(V_3f63911,X))) => s(fun(V_3f63911,V_3f63907),F) = s(fun(V_3f63911,V_3f63907),G))).
fof('h4/thm/nets/LIM_TENDS2_', conjecture, ![A,B]: ![M1, M2, F, X0, Y0]: (p(s(bool,'h4/const/topology/limpt'(s('h4/type/topology/topology'(A),'h4/const/topology/mtop'(s('h4/type/topology/metric'(A),M1))),s(A,X0),s(fun(A,bool),'h4/const/pred_set/UNIV')))) => (p(s(bool,'h4/const/nets/tends'(s(fun(A,B),F),s(B,Y0),s('h4/type/pair/prod'('h4/type/topology/topology'(B),fun(A,fun(A,bool))),'h4/const/pair/,'(s('h4/type/topology/topology'(B),'h4/const/topology/mtop'(s('h4/type/topology/metric'(B),M2))),s(fun(A,fun(A,bool)),'h4/const/nets/tendsto'(s('h4/type/pair/prod'('h4/type/topology/metric'(A),A),'h4/const/pair/,'(s('h4/type/topology/metric'(A),M1),s(A,X0)))))))))) <=> ![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)))) => ?[D]: (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',D)))) & ![X]: ((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','h4/const/topology/dist'(s('h4/type/topology/metric'(A),M1),s('h4/type/pair/prod'(A,A),'h4/const/pair/,'(s(A,X),s(A,X0)))))))) & p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real','h4/const/topology/dist'(s('h4/type/topology/metric'(A),M1),s('h4/type/pair/prod'(A,A),'h4/const/pair/,'(s(A,X),s(A,X0))))),s('h4/type/realax/real',D))))) => p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real','h4/const/topology/dist'(s('h4/type/topology/metric'(B),M2),s('h4/type/pair/prod'(B,B),'h4/const/pair/,'(s(B,happ(s(fun(A,B),F),s(A,X))),s(B,Y0))))),s('h4/type/realax/real',E)))))))))).
