%   ORIGINAL: 'h4/thm/nets/NET_NULL_CMUL_'
% 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: !g k x. 'h4/const/nets/tends' x ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/pair/,' ('h4/const/topology/mtop' 'h4/const/topology/mr1') g) ==> 'h4/const/nets/tends' (\n. 'h4/const/realax/real_mul' k (x n)) ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/pair/,' ('h4/const/topology/mtop' 'h4/const/topology/mr1') g)
%   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: !_0. (!k x n. happ (happ (happ _0 k) x) n = 'h4/const/realax/real_mul' k (happ x n)) ==> (!g k x. 'h4/const/nets/tends' x ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/pair/,' ('h4/const/topology/mtop' 'h4/const/topology/mr1') g) ==> 'h4/const/nets/tends' (happ (happ _0 k) x) ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/pair/,' ('h4/const/topology/mtop' 'h4/const/topology/mr1') g))
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_3f64111,V_3f64107]: ![F, G]: (![X]: s(V_3f64107,happ(s(fun(V_3f64111,V_3f64107),F),s(V_3f64111,X))) = s(V_3f64107,happ(s(fun(V_3f64111,V_3f64107),G),s(V_3f64111,X))) => s(fun(V_3f64111,V_3f64107),F) = s(fun(V_3f64111,V_3f64107),G))).
fof('h4/thm/nets/NET_NULL_CMUL_', conjecture, ![A]: ![V__0]: (![K, X, N]: s('h4/type/realax/real',happ(s(fun(A,'h4/type/realax/real'),happ(s(fun(fun(A,'h4/type/realax/real'),fun(A,'h4/type/realax/real')),happ(s(fun('h4/type/realax/real',fun(fun(A,'h4/type/realax/real'),fun(A,'h4/type/realax/real'))),V__0),s('h4/type/realax/real',K))),s(fun(A,'h4/type/realax/real'),X))),s(A,N))) = s('h4/type/realax/real','h4/const/realax/real_mul'(s('h4/type/realax/real',K),s('h4/type/realax/real',happ(s(fun(A,'h4/type/realax/real'),X),s(A,N))))) => ![G, K, X]: (p(s(bool,'h4/const/nets/tends'(s(fun(A,'h4/type/realax/real'),X),s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))),s('h4/type/pair/prod'('h4/type/topology/topology'('h4/type/realax/real'),fun(A,fun(A,bool))),'h4/const/pair/,'(s('h4/type/topology/topology'('h4/type/realax/real'),'h4/const/topology/mtop'(s('h4/type/topology/metric'('h4/type/realax/real'),'h4/const/topology/mr1'))),s(fun(A,fun(A,bool)),G)))))) => p(s(bool,'h4/const/nets/tends'(s(fun(A,'h4/type/realax/real'),happ(s(fun(fun(A,'h4/type/realax/real'),fun(A,'h4/type/realax/real')),happ(s(fun('h4/type/realax/real',fun(fun(A,'h4/type/realax/real'),fun(A,'h4/type/realax/real'))),V__0),s('h4/type/realax/real',K))),s(fun(A,'h4/type/realax/real'),X))),s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))),s('h4/type/pair/prod'('h4/type/topology/topology'('h4/type/realax/real'),fun(A,fun(A,bool))),'h4/const/pair/,'(s('h4/type/topology/topology'('h4/type/realax/real'),'h4/const/topology/mtop'(s('h4/type/topology/metric'('h4/type/realax/real'),'h4/const/topology/mr1'))),s(fun(A,fun(A,bool)),G))))))))).
