%   ORIGINAL: 'h4/thm/list/SINGL_SINGL_APPLY_'
% 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 f. 'h4/const/list/LIST_APPLY' ('h4/const/list/CONS' f 'h4/const/list/NIL') ('h4/const/list/CONS' x 'h4/const/list/NIL') = 'h4/const/list/CONS' (f x) 'h4/const/list/NIL'
%   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 f. 'h4/const/list/LIST_APPLY' ('h4/const/list/CONS' f 'h4/const/list/NIL') ('h4/const/list/CONS' x 'h4/const/list/NIL') = 'h4/const/list/CONS' (happ f x) 'h4/const/list/NIL'
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_3f11044,V_3f11040]: ![F, G]: (![X]: s(V_3f11040,happ(s(fun(V_3f11044,V_3f11040),F),s(V_3f11044,X))) = s(V_3f11040,happ(s(fun(V_3f11044,V_3f11040),G),s(V_3f11044,X))) => s(fun(V_3f11044,V_3f11040),F) = s(fun(V_3f11044,V_3f11040),G))).
fof('h4/thm/list/SINGL_SINGL_APPLY_', conjecture, ![B,A]: ![X, F]: s('h4/type/list/list'(A),'h4/const/list/LIST_APPLY'(s('h4/type/list/list'(fun(B,A)),'h4/const/list/CONS'(s(fun(B,A),F),s('h4/type/list/list'(fun(B,A)),'h4/const/list/NIL'))),s('h4/type/list/list'(B),'h4/const/list/CONS'(s(B,X),s('h4/type/list/list'(B),'h4/const/list/NIL'))))) = s('h4/type/list/list'(A),'h4/const/list/CONS'(s(A,happ(s(fun(B,A),F),s(B,X))),s('h4/type/list/list'(A),'h4/const/list/NIL')))).
