%   ORIGINAL: 'h4/thm/list/IMAGE_EL_count_LENGTH_'
% 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: !f ls. 'h4/const/pred_set/IMAGE' (\n. f ('h4/const/list/EL' n ls)) ('h4/const/pred_set/count' ('h4/const/list/LENGTH' ls)) = 'h4/const/pred_set/IMAGE' f ('h4/const/list/LIST_TO_SET' ls)
%   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. (!f ls n. happ (happ (happ _0 f) ls) n = happ f ('h4/const/list/EL' n ls)) ==> (!f ls. 'h4/const/pred_set/IMAGE' (happ (happ _0 f) ls) ('h4/const/pred_set/count' ('h4/const/list/LENGTH' ls)) = 'h4/const/pred_set/IMAGE' f ('h4/const/list/LIST_TO_SET' ls))
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_3f11994,V_3f11990]: ![F, G]: (![X]: s(V_3f11990,happ(s(fun(V_3f11994,V_3f11990),F),s(V_3f11994,X))) = s(V_3f11990,happ(s(fun(V_3f11994,V_3f11990),G),s(V_3f11994,X))) => s(fun(V_3f11994,V_3f11990),F) = s(fun(V_3f11994,V_3f11990),G))).
fof('h4/thm/list/IMAGE_EL_count_LENGTH_', conjecture, ![B,A]: ![V__0]: (![F, Ls, N]: s(B,happ(s(fun('h4/type/num/num',B),happ(s(fun('h4/type/list/list'(A),fun('h4/type/num/num',B)),happ(s(fun(fun(A,B),fun('h4/type/list/list'(A),fun('h4/type/num/num',B))),V__0),s(fun(A,B),F))),s('h4/type/list/list'(A),Ls))),s('h4/type/num/num',N))) = s(B,happ(s(fun(A,B),F),s(A,'h4/const/list/EL'(s('h4/type/num/num',N),s('h4/type/list/list'(A),Ls))))) => ![F, Ls]: s(fun(B,bool),'h4/const/pred_set/IMAGE'(s(fun('h4/type/num/num',B),happ(s(fun('h4/type/list/list'(A),fun('h4/type/num/num',B)),happ(s(fun(fun(A,B),fun('h4/type/list/list'(A),fun('h4/type/num/num',B))),V__0),s(fun(A,B),F))),s('h4/type/list/list'(A),Ls))),s(fun('h4/type/num/num',bool),'h4/const/pred_set/count'(s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),Ls))))))) = s(fun(B,bool),'h4/const/pred_set/IMAGE'(s(fun(A,B),F),s(fun(A,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(A),Ls))))))).
