%   ORIGINAL: 'h4/thm/hreal/HREAL_SUP_ISACUT_'
% 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) /\ (?Y. !X. P X ==> 'h4/const/hreal/hreal_lt' X Y) ==> 'h4/const/hreal/isacut' (\w. ?X. P X /\ 'h4/const/hreal/cut' X w)
%   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: !_0. (!P w. happ (happ _0 P) w <=> (?X. happ P X /\ 'h4/const/hreal/cut' X w)) ==> (!P. (?X. happ P X) /\ (?Y. !X. happ P X ==> 'h4/const/hreal/hreal_lt' X Y) ==> 'h4/const/hreal/isacut' (happ _0 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_3f37329,V_3f37325]: ![F, G]: (![X]: s(V_3f37325,happ(s(fun(V_3f37329,V_3f37325),F),s(V_3f37329,X))) = s(V_3f37325,happ(s(fun(V_3f37329,V_3f37325),G),s(V_3f37329,X))) => s(fun(V_3f37329,V_3f37325),F) = s(fun(V_3f37329,V_3f37325),G))).
fof('h4/thm/hreal/HREAL_SUP_ISACUT_', conjecture, ![V__0]: (![P, W]: (p(s(bool,happ(s(fun('h4/type/hrat/hrat',bool),happ(s(fun(fun('h4/type/hreal/hreal',bool),fun('h4/type/hrat/hrat',bool)),V__0),s(fun('h4/type/hreal/hreal',bool),P))),s('h4/type/hrat/hrat',W)))) <=> ?[X]: (p(s(bool,happ(s(fun('h4/type/hreal/hreal',bool),P),s('h4/type/hreal/hreal',X)))) & p(s(bool,'h4/const/hreal/cut'(s('h4/type/hreal/hreal',X),s('h4/type/hrat/hrat',W)))))) => ![P]: ((?[X]: p(s(bool,happ(s(fun('h4/type/hreal/hreal',bool),P),s('h4/type/hreal/hreal',X)))) & ?[Y]: ![X]: (p(s(bool,happ(s(fun('h4/type/hreal/hreal',bool),P),s('h4/type/hreal/hreal',X)))) => p(s(bool,'h4/const/hreal/hreal_lt'(s('h4/type/hreal/hreal',X),s('h4/type/hreal/hreal',Y)))))) => p(s(bool,'h4/const/hreal/isacut'(s(fun('h4/type/hrat/hrat',bool),happ(s(fun(fun('h4/type/hreal/hreal',bool),fun('h4/type/hrat/hrat',bool)),V__0),s(fun('h4/type/hreal/hreal',bool),P))))))))).
