%   ORIGINAL: 'h4/thm/pred_set/FINITE_INDUCT_'
% 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: !P. P 'h4/const/pred_set/EMPTY' /\ (!s. 'h4/const/pred_set/FINITE' s /\ P s ==> (!e. ~'h4/const/bool/IN' e s ==> P ('h4/const/pred_set/INSERT' e s))) ==> (!s. 'h4/const/pred_set/FINITE' s ==> P s)
%   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: !P. happ P 'h4/const/pred_set/EMPTY' /\ (!s. 'h4/const/pred_set/FINITE' s /\ happ P s ==> (!e. ~'h4/const/bool/IN' e s ==> happ P ('h4/const/pred_set/INSERT' e s))) ==> (!s. 'h4/const/pred_set/FINITE' s ==> happ P s)
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_3f19045,V_3f19041]: ![F, G]: (![X]: s(V_3f19041,happ(s(fun(V_3f19045,V_3f19041),F),s(V_3f19045,X))) = s(V_3f19041,happ(s(fun(V_3f19045,V_3f19041),G),s(V_3f19045,X))) => s(fun(V_3f19045,V_3f19041),F) = s(fun(V_3f19045,V_3f19041),G))).
fof('h4/thm/pred_set/FINITE_INDUCT_', conjecture, ![A]: ![P]: ((p(s(bool,happ(s(fun(fun(A,bool),bool),P),s(fun(A,bool),'h4/const/pred_set/EMPTY')))) & ![S]: ((p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) & p(s(bool,happ(s(fun(fun(A,bool),bool),P),s(fun(A,bool),S))))) => ![E]: (~ (p(s(bool,'h4/const/bool/IN'(s(A,E),s(fun(A,bool),S))))) => p(s(bool,happ(s(fun(fun(A,bool),bool),P),s(fun(A,bool),'h4/const/pred_set/INSERT'(s(A,E),s(fun(A,bool),S))))))))) => ![S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) => p(s(bool,happ(s(fun(fun(A,bool),bool),P),s(fun(A,bool),S))))))).
