%   ORIGINAL: h4/real/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
% Assm: h4/realax/REAL__SUP__ALLPOS: !P. (!x. P x ==> h4/realax/real__lt h4/realax/real__0 x) /\ (?x. P x) /\ (?z. !x. P x ==> h4/realax/real__lt x z) ==> (?s. !y. (?x. P x /\ h4/realax/real__lt y x) <=> h4/realax/real__lt y s)
% Assm: h4/real/REAL__0: h4/realax/real__0 = h4/real/real__of__num h4/num/0
% Goal: !P. (!x. P x ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) /\ (?x. P x) /\ (?z. !x. P x ==> h4/realax/real__lt x z) ==> (?s. !y. (?x. P x /\ h4/realax/real__lt y x) <=> h4/realax/real__lt y s)
%   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_realaxs_REALu_u_SUPu_u_ALLPOS]: !P. (!x. happ P x ==> h4/realax/real__lt h4/realax/real__0 x) /\ (?x. happ P x) /\ (?z. !x. happ P x ==> h4/realax/real__lt x z) ==> (?s. !y. (?x. happ P x /\ h4/realax/real__lt y x) <=> h4/realax/real__lt y s)
% Assm [h4s_reals_REALu_u_0]: h4/realax/real__0 = h4/real/real__of__num h4/num/0
% Goal: !P. (!x. happ P x ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) /\ (?x. happ P x) /\ (?z. !x. happ P x ==> h4/realax/real__lt x z) ==> (?s. !y. (?x. happ P x /\ h4/realax/real__lt y x) <=> h4/realax/real__lt y s)
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_Q60326,TV_Q60322]: ![V_f, V_g]: (![V_x]: s(TV_Q60322,happ(s(t_fun(TV_Q60326,TV_Q60322),V_f),s(TV_Q60326,V_x))) = s(TV_Q60322,happ(s(t_fun(TV_Q60326,TV_Q60322),V_g),s(TV_Q60326,V_x))) => s(t_fun(TV_Q60326,TV_Q60322),V_f) = s(t_fun(TV_Q60326,TV_Q60322),V_g))).
fof(ah4s_realaxs_REALu_u_SUPu_u_ALLPOS, axiom, ![V_P]: ((![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_0),s(t_h4s_realaxs_real,V_x))))) & (?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) & ?[V_z]: ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z))))))) => ?[V_s]: ![V_y]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))) <=> p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_s))))))).
fof(ah4s_reals_REALu_u_0, axiom, s(t_h4s_realaxs_real,h4s_realaxs_realu_u_0) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ch4s_reals_REALu_u_SUPu_u_ALLPOS, conjecture, ![V_P]: ((![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))))) & (?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) & ?[V_z]: ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z))))))) => ?[V_s]: ![V_y]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))) <=> p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_s))))))).
