%   ORIGINAL: 'h4/thm/pred_set/SUBSET_INSERT_'
% 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 s. ~'h4/const/bool/IN' x s ==> (!t. 'h4/const/pred_set/SUBSET' s ('h4/const/pred_set/INSERT' x t) <=> 'h4/const/pred_set/SUBSET' s t)
%   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 s. ~'h4/const/bool/IN' x s ==> (!t. 'h4/const/pred_set/SUBSET' s ('h4/const/pred_set/INSERT' x t) <=> 'h4/const/pred_set/SUBSET' s t)
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_3f17044,V_3f17040]: ![F, G]: (![X]: s(V_3f17040,happ(s(fun(V_3f17044,V_3f17040),F),s(V_3f17044,X))) = s(V_3f17040,happ(s(fun(V_3f17044,V_3f17040),G),s(V_3f17044,X))) => s(fun(V_3f17044,V_3f17040),F) = s(fun(V_3f17044,V_3f17040),G))).
fof('h4/thm/pred_set/SUBSET_INSERT_', conjecture, ![A]: ![X, S]: (~ (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S))))) => ![T]: s(bool,'h4/const/pred_set/SUBSET'(s(fun(A,bool),S),s(fun(A,bool),'h4/const/pred_set/INSERT'(s(A,X),s(fun(A,bool),T))))) = s(bool,'h4/const/pred_set/SUBSET'(s(fun(A,bool),S),s(fun(A,bool),T))))).
