%   ORIGINAL: 'h4/thm/real/REAL_INF_LE_'
% 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 x. (?y. p y) /\ (?y. !z. p z ==> 'h4/const/real/real_lte' y z) ==> ('h4/const/real/real_lte' ('h4/const/real/inf' p) x <=> (!y. (!z. p z ==> 'h4/const/real/real_lte' y z) ==> 'h4/const/real/real_lte' y x))
%   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 x. (?y. happ p y) /\ (?y. !z. happ p z ==> 'h4/const/real/real_lte' y z) ==> ('h4/const/real/real_lte' ('h4/const/real/inf' p) x <=> (!y. (!z. happ p z ==> 'h4/const/real/real_lte' y z) ==> 'h4/const/real/real_lte' y x))
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_3f96745,V_3f96741]: ![F, G]: (![X]: s(V_3f96741,happ(s(fun(V_3f96745,V_3f96741),F),s(V_3f96745,X))) = s(V_3f96741,happ(s(fun(V_3f96745,V_3f96741),G),s(V_3f96745,X))) => s(fun(V_3f96745,V_3f96741),F) = s(fun(V_3f96745,V_3f96741),G))).
fof('h4/thm/real/REAL_INF_LE_', conjecture, ![P, X]: ((?[Y]: p(s(bool,happ(s(fun('h4/type/realax/real',bool),P),s('h4/type/realax/real',Y)))) & ?[Y]: ![Z]: (p(s(bool,happ(s(fun('h4/type/realax/real',bool),P),s('h4/type/realax/real',Z)))) => p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',Y),s('h4/type/realax/real',Z)))))) => (p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real','h4/const/real/inf'(s(fun('h4/type/realax/real',bool),P))),s('h4/type/realax/real',X)))) <=> ![Y]: (![Z]: (p(s(bool,happ(s(fun('h4/type/realax/real',bool),P),s('h4/type/realax/real',Z)))) => p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',Y),s('h4/type/realax/real',Z))))) => p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',Y),s('h4/type/realax/real',X)))))))).
