%   ORIGINAL: 'h4/thm/list/MEM_FLAT_'
% 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: !x L. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ('h4/const/list/FLAT' L)) <=> (?l. 'h4/const/bool/IN' l ('h4/const/list/LIST_TO_SET' L) /\ 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' l))
%   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: !x L. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ('h4/const/list/FLAT' L)) <=> (?l. 'h4/const/bool/IN' l ('h4/const/list/LIST_TO_SET' L) /\ 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' l))
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_3f3813,V_3f3809]: ![F, G]: (![X]: s(V_3f3809,happ(s(fun(V_3f3813,V_3f3809),F),s(V_3f3813,X))) = s(V_3f3809,happ(s(fun(V_3f3813,V_3f3809),G),s(V_3f3813,X))) => s(fun(V_3f3813,V_3f3809),F) = s(fun(V_3f3813,V_3f3809),G))).
fof('h4/thm/list/MEM_FLAT_', conjecture, ![A]: ![X, L]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(A),'h4/const/list/FLAT'(s('h4/type/list/list'('h4/type/list/list'(A)),L)))))))) <=> ?[L0]: (p(s(bool,'h4/const/bool/IN'(s('h4/type/list/list'(A),L0),s(fun('h4/type/list/list'(A),bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'('h4/type/list/list'(A)),L)))))) & p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(A),L0))))))))).
