%   ORIGINAL: 'h4/thm/list/SUM_eq_0_'
% 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: !ls. 'h4/const/list/SUM' ls = 'h4/const/num/0' <=> (!x. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ls) ==> x = 'h4/const/num/0')
%   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: !ls. 'h4/const/list/SUM' ls = 'h4/const/num/0' <=> (!x. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ls) ==> x = 'h4/const/num/0')
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_3f5414,V_3f5410]: ![F, G]: (![X]: s(V_3f5410,happ(s(fun(V_3f5414,V_3f5410),F),s(V_3f5414,X))) = s(V_3f5410,happ(s(fun(V_3f5414,V_3f5410),G),s(V_3f5414,X))) => s(fun(V_3f5414,V_3f5410),F) = s(fun(V_3f5414,V_3f5410),G))).
fof('h4/thm/list/SUM_eq_0_', conjecture, ![Ls]: (s('h4/type/num/num','h4/const/list/SUM'(s('h4/type/list/list'('h4/type/num/num'),Ls))) = s('h4/type/num/num','h4/const/num/0') <=> ![X]: (p(s(bool,'h4/const/bool/IN'(s('h4/type/num/num',X),s(fun('h4/type/num/num',bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'('h4/type/num/num'),Ls)))))) => s('h4/type/num/num',X) = s('h4/type/num/num','h4/const/num/0')))).
