%   ORIGINAL: 'h4/thm/pred_set/MIN_SET_ELIM_'
% 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 Q. ~(P = 'h4/const/pred_set/EMPTY') /\ (!x. (!y. 'h4/const/bool/IN' y P ==> 'h4/const/arithmetic/<=' x y) /\ 'h4/const/bool/IN' x P ==> Q x) ==> Q ('h4/const/pred_set/MIN_SET' P)
%   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 Q. ~(P = 'h4/const/pred_set/EMPTY') /\ (!x. (!y. 'h4/const/bool/IN' y P ==> 'h4/const/arithmetic/<=' x y) /\ 'h4/const/bool/IN' x P ==> happ Q x) ==> happ Q ('h4/const/pred_set/MIN_SET' P)
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_3f22973,V_3f22969]: ![F, G]: (![X]: s(V_3f22969,happ(s(fun(V_3f22973,V_3f22969),F),s(V_3f22973,X))) = s(V_3f22969,happ(s(fun(V_3f22973,V_3f22969),G),s(V_3f22973,X))) => s(fun(V_3f22973,V_3f22969),F) = s(fun(V_3f22973,V_3f22969),G))).
fof('h4/thm/pred_set/MIN_SET_ELIM_', conjecture, ![P, Q]: ((~ (s(fun('h4/type/num/num',bool),P) = s(fun('h4/type/num/num',bool),'h4/const/pred_set/EMPTY')) & ![X]: ((![Y]: (p(s(bool,'h4/const/bool/IN'(s('h4/type/num/num',Y),s(fun('h4/type/num/num',bool),P)))) => p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num',X),s('h4/type/num/num',Y))))) & p(s(bool,'h4/const/bool/IN'(s('h4/type/num/num',X),s(fun('h4/type/num/num',bool),P))))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),Q),s('h4/type/num/num',X)))))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),Q),s('h4/type/num/num','h4/const/pred_set/MIN_SET'(s(fun('h4/type/num/num',bool),P)))))))).
