%   ORIGINAL: 'h4/thm/rich_list/ELL_MEM_'
% 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: !n l. 'h4/const/prim_rec/<' n ('h4/const/list/LENGTH' l) ==> 'h4/const/bool/IN' ('h4/const/rich_list/ELL' n l) ('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: !n l. 'h4/const/prim_rec/<' n ('h4/const/list/LENGTH' l) ==> 'h4/const/bool/IN' ('h4/const/rich_list/ELL' n l) ('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_3f56161,V_3f56157]: ![F, G]: (![X]: s(V_3f56157,happ(s(fun(V_3f56161,V_3f56157),F),s(V_3f56161,X))) = s(V_3f56157,happ(s(fun(V_3f56161,V_3f56157),G),s(V_3f56161,X))) => s(fun(V_3f56161,V_3f56157),F) = s(fun(V_3f56161,V_3f56157),G))).
fof('h4/thm/rich_list/ELL_MEM_', conjecture, ![A]: ![N, L]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),L)))))) => p(s(bool,'h4/const/bool/IN'(s(A,'h4/const/rich_list/ELL'(s('h4/type/num/num',N),s('h4/type/list/list'(A),L))),s(fun(A,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(A),L)))))))).
