%   ORIGINAL: 'h4/thm/topology/METRIC_ZERO_'
% 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 y. 'h4/const/topology/dist' m ('h4/const/pair/,' x y) = 'h4/const/real/real_of_num' 'h4/const/num/0' <=> x = 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: !m x y. 'h4/const/topology/dist' m ('h4/const/pair/,' x y) = 'h4/const/real/real_of_num' 'h4/const/num/0' <=> x = 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_3f37705,V_3f37701]: ![F, G]: (![X]: s(V_3f37701,happ(s(fun(V_3f37705,V_3f37701),F),s(V_3f37705,X))) = s(V_3f37701,happ(s(fun(V_3f37705,V_3f37701),G),s(V_3f37705,X))) => s(fun(V_3f37705,V_3f37701),F) = s(fun(V_3f37705,V_3f37701),G))).
fof('h4/thm/topology/METRIC_ZERO_', conjecture, ![A]: ![M, X, Y]: (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','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))) <=> s(A,X) = s(A,Y))).
