%   ORIGINAL: 'h4/thm/realax/REAL_SUP_ALLPOS_'
% 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 ==> 'h4/const/realax/real_lt' 'h4/const/realax/real_0' x) /\ (?x. P x) /\ (?z. !x. P x ==> 'h4/const/realax/real_lt' x z) ==> (?s. !y. (?x. P x /\ 'h4/const/realax/real_lt' y x) <=> 'h4/const/realax/real_lt' y 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. (!x. happ P x ==> 'h4/const/realax/real_lt' 'h4/const/realax/real_0' x) /\ (?x. happ P x) /\ (?z. !x. happ P x ==> 'h4/const/realax/real_lt' x z) ==> (?s. !y. (?x. happ P x /\ 'h4/const/realax/real_lt' y x) <=> 'h4/const/realax/real_lt' y 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_3f50111,V_3f50107]: ![F, G]: (![X]: s(V_3f50107,happ(s(fun(V_3f50111,V_3f50107),F),s(V_3f50111,X))) = s(V_3f50107,happ(s(fun(V_3f50111,V_3f50107),G),s(V_3f50111,X))) => s(fun(V_3f50111,V_3f50107),F) = s(fun(V_3f50111,V_3f50107),G))).
fof('h4/thm/realax/REAL_SUP_ALLPOS_', conjecture, ![P]: ((![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','h4/const/realax/real_0'),s('h4/type/realax/real',X))))) & (?[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))))))) => ?[S]: ![Y]: (?[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',Y),s('h4/type/realax/real',X))))) <=> p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real',Y),s('h4/type/realax/real',S))))))).
