%   ORIGINAL: h4/hreal/HREAL__SUP
% 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
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/hreal/hreal__tybij_c1: !r. h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm: h4/hreal/hreal__sup0: !P. h4/hreal/hreal__sup P = h4/hreal/hreal0 (\w. ?X. P X /\ h4/hreal/cut X w)
% Assm: h4/hreal/hreal__lt0: !Y X. h4/hreal/hreal__lt X Y <=> ~(X = Y) /\ (!x. h4/hreal/cut X x ==> h4/hreal/cut Y x)
% Assm: h4/hreal/HREAL__LT__LEMMA: !Y X. h4/hreal/hreal__lt X Y ==> (?x. ~h4/hreal/cut X x /\ h4/hreal/cut Y x)
% Assm: h4/hreal/HREAL__LT__TOTAL: !Y X. X = Y \/ h4/hreal/hreal__lt X Y \/ h4/hreal/hreal__lt Y X
% Assm: h4/hreal/HREAL__SUP__ISACUT: !P. (?X. P X) /\ (?Y. !X. P X ==> h4/hreal/hreal__lt X Y) ==> h4/hreal/isacut (\w. ?X. P X /\ h4/hreal/cut X w)
% Goal: !P. (?X. P X) /\ (?Y. !X. P X ==> h4/hreal/hreal__lt X Y) ==> (!Y. (?X. P X /\ h4/hreal/hreal__lt Y X) <=> h4/hreal/hreal__lt Y (h4/hreal/hreal__sup P))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_hreals_hrealu_u_tybiju_c1]: !r. h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm [h4s_hreals_hrealu_u_sup0]: !_0. (!P w. happ (happ _0 P) w <=> (?X. happ P X /\ happ (h4/hreal/cut X) w)) ==> (!P. h4/hreal/hreal__sup P = h4/hreal/hreal0 (happ _0 P))
% Assm [h4s_hreals_hrealu_u_lt0]: !Y X. h4/hreal/hreal__lt X Y <=> ~(X = Y) /\ (!x. happ (h4/hreal/cut X) x ==> happ (h4/hreal/cut Y) x)
% Assm [h4s_hreals_HREALu_u_LTu_u_LEMMA]: !Y X. h4/hreal/hreal__lt X Y ==> (?x. ~happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) x)
% Assm [h4s_hreals_HREALu_u_LTu_u_TOTAL]: !Y X. X = Y \/ h4/hreal/hreal__lt X Y \/ h4/hreal/hreal__lt Y X
% Assm [h4s_hreals_HREALu_u_SUPu_u_ISACUT]: !_0. (!P w. happ (happ _0 P) w <=> (?X. happ P X /\ happ (h4/hreal/cut X) w)) ==> (!P. (?X. happ P X) /\ (?Y. !X. happ P X ==> h4/hreal/hreal__lt X Y) ==> h4/hreal/isacut (happ _0 P))
% Goal: !P. (?X. happ P X) /\ (?Y. !X. happ P X ==> h4/hreal/hreal__lt X Y) ==> (!Y. (?X. happ P X /\ h4/hreal/hreal__lt Y X) <=> h4/hreal/hreal__lt Y (h4/hreal/hreal__sup P))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q255081,TV_Q255077]: ![V_f, V_g]: (![V_x]: s(TV_Q255077,happ(s(t_fun(TV_Q255081,TV_Q255077),V_f),s(TV_Q255081,V_x))) = s(TV_Q255077,happ(s(t_fun(TV_Q255081,TV_Q255077),V_g),s(TV_Q255081,V_x))) => s(t_fun(TV_Q255081,TV_Q255077),V_f) = s(t_fun(TV_Q255081,TV_Q255077),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_hreals_hrealu_u_tybiju_c1, axiom, ![V_r]: (p(s(t_bool,h4s_hreals_isacut(s(t_fun(t_h4s_hrats_hrat,t_bool),V_r)))) <=> s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))))) = s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))).
fof(ah4s_hreals_hrealu_u_sup0, axiom, ![V_uu_0]: (![V_P, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))))) => ![V_P]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sup(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))))))).
fof(ah4s_hreals_hrealu_u_lt0, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) <=> (~ (s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y)) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_x)))))))).
fof(ah4s_hreals_HREALu_u_LTu_u_LEMMA, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) => ?[V_x]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_x))))))).
fof(ah4s_hreals_HREALu_u_LTu_u_TOTAL, axiom, ![V_Y, V_X]: (s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y) | (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) | p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X))))))).
fof(ah4s_hreals_HREALu_u_SUPu_u_ISACUT, axiom, ![V_uu_0]: (![V_P, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))))) => ![V_P]: ((?[V_X]: p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & ?[V_Y]: ![V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) => p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))))) => p(s(t_bool,h4s_hreals_isacut(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))))))))).
fof(ch4s_hreals_HREALu_u_SUP, conjecture, ![V_P]: ((?[V_X]: p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & ?[V_Y]: ![V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) => p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))))) => ![V_Y]: (?[V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X))))) <=> p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sup(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))))))))).
