%   ORIGINAL: 'h4/thm/real/SETOK_LE_LT_'
% 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. P x) /\ (?z. !x. P x ==> 'h4/const/real/real_lte' x z) <=> (?x. P x) /\ (?z. !x. P x ==> 'h4/const/realax/real_lt' x z)
%   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. happ P x) /\ (?z. !x. happ P x ==> 'h4/const/real/real_lte' x z) <=> (?x. happ P x) /\ (?z. !x. happ P x ==> 'h4/const/realax/real_lt' x z)
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_3f93819,V_3f93815]: ![F, G]: (![X]: s(V_3f93815,happ(s(fun(V_3f93819,V_3f93815),F),s(V_3f93819,X))) = s(V_3f93815,happ(s(fun(V_3f93819,V_3f93815),G),s(V_3f93819,X))) => s(fun(V_3f93819,V_3f93815),F) = s(fun(V_3f93819,V_3f93815),G))).
fof('h4/thm/real/SETOK_LE_LT_', conjecture, ![P]: ((?[X]: p(s(bool,happ(s(fun('h4/type/realax/real',bool),P),s('h4/type/realax/real',X)))) & ?[Z]: ![X]: (p(s(bool,happ(s(fun('h4/type/realax/real',bool),P),s('h4/type/realax/real',X)))) => p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',X),s('h4/type/realax/real',Z)))))) <=> (?[X]: p(s(bool,happ(s(fun('h4/type/realax/real',bool),P),s('h4/type/realax/real',X)))) & ?[Z]: ![X]: (p(s(bool,happ(s(fun('h4/type/realax/real',bool),P),s('h4/type/realax/real',X)))) => p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real',X),s('h4/type/realax/real',Z)))))))).
