TPTP Axioms File: NUM007^3.ax
%------------------------------------------------------------------------------
% File : NUM007^3 : TPTP v8.2.0. Released v7.1.0.
% Domain : Number Theory
% Axioms : Grundlagen chapter 3
% Version : [Bro17] axioms : Especial.
% English :
% Refs : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source : [Bro17]
% Names :
% Status : Satisfiable
% Syntax : Number of formulae : 378 ( 117 unt; 117 typ; 117 def)
% Number of atoms : 1926 ( 117 equ; 0 cnn)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 3227 ( 0 ~; 0 |; 0 &;3065 @)
% ( 0 <=>; 162 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg;3065 nst)
% Number of types : 2 ( 0 usr)
% Number of type conns : 209 ( 209 >; 0 *; 0 +; 0 <<)
% Number of symbols : 176 ( 175 usr; 68 con; 0-5 aty)
% Number of variables : 953 ( 950 ^ 3 !; 0 ?; 953 :)
% SPC :
% Comments :
%------------------------------------------------------------------------------
thf(typ_rp_is,type,
rp_is: $i > $i > $o ).
thf(def_rp_is,definition,
( rp_is
= ( e_is @ cut ) ) ).
thf(typ_rp_nis,type,
rp_nis: $i > $i > $o ).
thf(def_rp_nis,definition,
( rp_nis
= ( ^ [X0: $i,X1: $i] : ( d_not @ ( rp_is @ X0 @ X1 ) ) ) ) ).
thf(typ_cutof,type,
cutof: $i > $i ).
thf(def_cutof,definition,
( cutof
= ( out @ ( power @ rat ) @ cutprop ) ) ).
thf(typ_rp_all,type,
rp_all: ( $i > $o ) > $o ).
thf(def_rp_all,definition,
( rp_all
= ( all @ cut ) ) ).
thf(typ_rp_some,type,
rp_some: ( $i > $o ) > $o ).
thf(def_rp_some,definition,
( rp_some
= ( l_some @ cut ) ) ).
thf(typ_rp_one,type,
rp_one: ( $i > $o ) > $o ).
thf(def_rp_one,definition,
( rp_one
= ( one @ cut ) ) ).
thf(satz116,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] : ( rp_is @ X0 @ X0 ) ) ).
thf(satz117,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_is @ X0 @ X1 )
=> ( rp_is @ X1 @ X0 ) ) ) ) ).
thf(satz118,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_is @ X1 @ X2 )
=> ( rp_is @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz119,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( urt @ X0 @ X1 )
=> ( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( rt_more @ X2 @ X1 )
=> ( urt @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz119a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( urt @ X0 @ X1 )
=> ( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( rt_less @ X1 @ X2 )
=> ( urt @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz120,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( lrt @ X0 @ X1 )
=> ( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( rt_less @ X2 @ X1 )
=> ( lrt @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz120a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( lrt @ X0 @ X1 )
=> ( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( rt_more @ X1 @ X2 )
=> ( lrt @ X0 @ X2 ) ) ) ) ) ) ).
thf(typ_rp_more,type,
rp_more: $i > $i > $o ).
thf(def_rp_more,definition,
( rp_more
= ( ^ [X0: $i,X1: $i] :
( rt_some
@ ^ [X2: $i] : ( d_and @ ( lrt @ X0 @ X2 ) @ ( urt @ X1 @ X2 ) ) ) ) ) ).
thf(typ_rp_less,type,
rp_less: $i > $i > $o ).
thf(def_rp_less,definition,
( rp_less
= ( ^ [X0: $i,X1: $i] :
( rt_some
@ ^ [X2: $i] : ( d_and @ ( urt @ X0 @ X2 ) @ ( lrt @ X1 @ X2 ) ) ) ) ) ).
thf(satz121,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_less @ X1 @ X0 ) ) ) ) ).
thf(satz122,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_less @ X0 @ X1 )
=> ( rp_more @ X1 @ X0 ) ) ) ) ).
thf(satz123,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( orec3 @ ( rp_is @ X0 @ X1 ) @ ( rp_more @ X0 @ X1 ) @ ( rp_less @ X0 @ X1 ) ) ) ) ).
thf(k_satz123a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( or3 @ ( rp_is @ X0 @ X1 ) @ ( rp_more @ X0 @ X1 ) @ ( rp_less @ X0 @ X1 ) ) ) ) ).
thf(k_satz123b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( ec3 @ ( rp_is @ X0 @ X1 ) @ ( rp_more @ X0 @ X1 ) @ ( rp_less @ X0 @ X1 ) ) ) ) ).
thf(typ_rp_moreis,type,
rp_moreis: $i > $i > $o ).
thf(def_rp_moreis,definition,
( rp_moreis
= ( ^ [X0: $i,X1: $i] : ( l_or @ ( rp_more @ X0 @ X1 ) @ ( rp_is @ X0 @ X1 ) ) ) ) ).
thf(typ_rp_lessis,type,
rp_lessis: $i > $i > $o ).
thf(def_rp_lessis,definition,
( rp_lessis
= ( ^ [X0: $i,X1: $i] : ( l_or @ ( rp_less @ X0 @ X1 ) @ ( rp_is @ X0 @ X1 ) ) ) ) ).
thf(satz124,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_moreis @ X0 @ X1 )
=> ( rp_lessis @ X1 @ X0 ) ) ) ) ).
thf(satz125,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( rp_moreis @ X1 @ X0 ) ) ) ) ).
thf(satz123c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_moreis @ X0 @ X1 )
=> ( d_not @ ( rp_less @ X0 @ X1 ) ) ) ) ) ).
thf(satz123d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( d_not @ ( rp_more @ X0 @ X1 ) ) ) ) ) ).
thf(satz123e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( d_not @ ( rp_more @ X0 @ X1 ) )
=> ( rp_lessis @ X0 @ X1 ) ) ) ) ).
thf(satz123f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( d_not @ ( rp_less @ X0 @ X1 ) )
=> ( rp_moreis @ X0 @ X1 ) ) ) ) ).
thf(satz123g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( d_not @ ( rp_lessis @ X0 @ X1 ) ) ) ) ) ).
thf(satz123h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_less @ X0 @ X1 )
=> ( d_not @ ( rp_moreis @ X0 @ X1 ) ) ) ) ) ).
thf(satz123j,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( d_not @ ( rp_moreis @ X0 @ X1 ) )
=> ( rp_less @ X0 @ X1 ) ) ) ) ).
thf(satz123k,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( d_not @ ( rp_lessis @ X0 @ X1 ) )
=> ( rp_more @ X0 @ X1 ) ) ) ) ).
thf(satz126,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ X0 @ X1 )
=> ( ( rp_less @ X1 @ X2 )
=> ( rp_less @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz127a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( ( rp_less @ X1 @ X2 )
=> ( rp_less @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz127b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ X0 @ X1 )
=> ( ( rp_lessis @ X1 @ X2 )
=> ( rp_less @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz127c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_moreis @ X0 @ X1 )
=> ( ( rp_more @ X1 @ X2 )
=> ( rp_more @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz127d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ X0 @ X1 )
=> ( ( rp_moreis @ X1 @ X2 )
=> ( rp_more @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz128,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( ( rp_lessis @ X1 @ X2 )
=> ( rp_lessis @ X0 @ X2 ) ) ) ) ) ) ).
thf(typ_sumprop1,type,
sumprop1: $i > $i > $i > $i > $i > $o ).
thf(def_sumprop1,definition,
( sumprop1
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i] : ( and3 @ ( lrt @ X0 @ X3 ) @ ( lrt @ X1 @ X4 ) @ ( rt_is @ X2 @ ( rt_pl @ X3 @ X4 ) ) ) ) ) ).
thf(typ_sumprop,type,
sumprop: $i > $i > $i > $o ).
thf(def_sumprop,definition,
( sumprop
= ( ^ [X0: $i,X1: $i,X2: $i] :
( rt_some
@ ^ [X3: $i] : ( rt_some @ ( sumprop1 @ X0 @ X1 @ X2 @ X3 ) ) ) ) ) ).
thf(typ_sum,type,
sum: $i > $i > $i ).
thf(def_sum,definition,
( sum
= ( ^ [X0: $i,X1: $i] : ( d_Sep @ rat @ ( sumprop @ X0 @ X1 ) ) ) ) ).
thf(satz129a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( urt @ X0 @ X2 )
=> ( all_of
@ ^ [X3: $i] : ( in @ X3 @ rat )
@ ^ [X3: $i] :
( ( urt @ X1 @ X3 )
=> ( d_not @ ( rt_in @ ( rt_pl @ X2 @ X3 ) @ ( sum @ X0 @ X1 ) ) ) ) ) ) ) ) ) ).
thf(typ_d_3129_z1,type,
d_3129_z1: $i > $i > $i > $i ).
thf(def_d_3129_z1,definition,
( d_3129_z1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( rt_ov @ X0 @ ( rt_pl @ X1 @ X2 ) ) ) ) ).
thf(satz129,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( cutprop @ ( sum @ X0 @ X1 ) ) ) ) ).
thf(typ_rp_pl,type,
rp_pl: $i > $i > $i ).
thf(def_rp_pl,definition,
( rp_pl
= ( ^ [X0: $i,X1: $i] : ( cutof @ ( sum @ X0 @ X1 ) ) ) ) ).
thf(satz130,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_is @ ( rp_pl @ X0 @ X1 ) @ ( rp_pl @ X1 @ X0 ) ) ) ) ).
thf(satz131,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] : ( rp_is @ ( rp_pl @ ( rp_pl @ X0 @ X1 ) @ X2 ) @ ( rp_pl @ X0 @ ( rp_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(typ_d_3132_prop1,type,
d_3132_prop1: $i > $i > $i > $o ).
thf(def_d_3132_prop1,definition,
( d_3132_prop1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( d_and @ ( lrt @ X0 @ X1 ) @ ( urt @ X0 @ X2 ) ) ) ) ).
thf(typ_d_3132_prop2,type,
d_3132_prop2: $i > $i > $i > $i > $o ).
thf(def_d_3132_prop2,definition,
( d_3132_prop2
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] : ( rt_is @ ( rt_mn @ X3 @ X2 ) @ X1 ) ) ) ).
thf(typ_d_3132_prop3,type,
d_3132_prop3: $i > $i > $i > $i > $o ).
thf(def_d_3132_prop3,definition,
( d_3132_prop3
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
( d_and @ ( d_3132_prop1 @ X0 @ X2 @ X3 )
@ ( ( d_3132_prop1 @ X0 @ X2 @ X3 )
=> ( d_3132_prop2 @ X0 @ X1 @ X2 @ X3 ) ) ) ) ) ).
thf(typ_d_3132_prop4,type,
d_3132_prop4: $i > $i > $o ).
thf(def_d_3132_prop4,definition,
( d_3132_prop4
= ( ^ [X0: $i,X1: $i] :
( rt_some
@ ^ [X2: $i] : ( rt_some @ ( d_3132_prop3 @ X0 @ X1 @ X2 ) ) ) ) ) ).
thf(typ_d_3132_u0,type,
d_3132_u0: $i > $i > $i ).
thf(def_d_3132_u0,definition,
( d_3132_u0
= ( ^ [X0: $i,X1: $i] : ( rt_pl @ X1 @ X0 ) ) ) ).
thf(typ_um10,type,
um10: $i > $i ).
thf(def_um10,definition,
( um10
= ( ^ [X0: $i] : ( rt_mn @ ( rtofn @ X0 ) @ d_1rt ) ) ) ).
thf(typ_um1,type,
um1: $i > $i ).
thf(def_um1,definition,
( um1
= ( ^ [X0: $i] : ( nofrt @ ( um10 @ X0 ) ) ) ) ).
thf(typ_d_3132_v0,type,
d_3132_v0: $i > $i > $i > $i ).
thf(def_d_3132_v0,definition,
( d_3132_v0
= ( ^ [X0: $i,X1: $i,X2: $i] : ( rt_pl @ X1 @ ( rt_ts @ ( um10 @ X2 ) @ X0 ) ) ) ) ).
thf(typ_d_3132_w0,type,
d_3132_w0: $i > $i > $i > $i ).
thf(def_d_3132_w0,definition,
( d_3132_w0
= ( ^ [X0: $i,X1: $i,X2: $i] : ( rt_pl @ X1 @ ( rt_ts @ ( rtofn @ X2 ) @ X0 ) ) ) ) ).
thf(satz132,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( rt_some
@ ^ [X2: $i] :
( rt_some
@ ^ [X3: $i] :
( d_and @ ( d_and @ ( lrt @ X0 @ X2 ) @ ( urt @ X0 @ X3 ) )
@ ( ( d_and @ ( lrt @ X0 @ X2 ) @ ( urt @ X0 @ X3 ) )
=> ( rt_is @ ( rt_mn @ X3 @ X2 ) @ X1 ) ) ) ) ) ) ) ).
thf(satz132app,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
! [X1: $o] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( all_of
@ ^ [X3: $i] : ( in @ X3 @ rat )
@ ^ [X3: $i] :
( ( lrt @ X0 @ X3 )
=> ( all_of
@ ^ [X4: $i] : ( in @ X4 @ rat )
@ ^ [X4: $i] :
( ( urt @ X0 @ X4 )
=> ( ( rt_is @ ( rt_mn @ X4 @ X3 ) @ X2 )
=> X1 ) ) ) ) )
=> X1 ) ) ) ).
thf(satz133,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_more @ ( rp_pl @ X0 @ X1 ) @ X0 ) ) ) ).
thf(satz133a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_less @ X0 @ ( rp_pl @ X0 @ X1 ) ) ) ) ).
thf(satz134,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_more @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz135b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ X0 @ X1 )
=> ( rp_is @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz135c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ X0 @ X1 )
=> ( rp_less @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz135d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_more @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz135e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ X0 @ X1 )
=> ( rp_is @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz135f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ X0 @ X1 )
=> ( rp_less @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz135g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz135h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz135j,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz135k,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz136a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X2 ) )
=> ( rp_more @ X0 @ X1 ) ) ) ) ) ).
thf(satz136b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X2 ) )
=> ( rp_is @ X0 @ X1 ) ) ) ) ) ).
thf(satz136c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X2 ) )
=> ( rp_less @ X0 @ X1 ) ) ) ) ) ).
thf(satz136d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X2 @ X1 ) )
=> ( rp_more @ X0 @ X1 ) ) ) ) ) ).
thf(satz136e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X2 @ X1 ) )
=> ( rp_is @ X0 @ X1 ) ) ) ) ) ).
thf(satz136f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ ( rp_pl @ X2 @ X0 ) @ ( rp_pl @ X2 @ X1 ) )
=> ( rp_less @ X0 @ X1 ) ) ) ) ) ).
thf(satz137,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_more @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz137a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_less @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz138a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_moreis @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz138b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_more @ X0 @ X1 )
=> ( ( rp_moreis @ X2 @ X3 )
=> ( rp_more @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz138c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz138d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_less @ X0 @ X1 )
=> ( ( rp_lessis @ X2 @ X3 )
=> ( rp_less @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz139,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_moreis @ X0 @ X1 )
=> ( ( rp_moreis @ X2 @ X3 )
=> ( rp_moreis @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz139a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( ( rp_lessis @ X2 @ X3 )
=> ( rp_lessis @ ( rp_pl @ X0 @ X2 ) @ ( rp_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz140b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ ( rp_pl @ X1 @ X2 ) @ X0 )
=> ( ( rp_is @ ( rp_pl @ X1 @ X3 ) @ X0 )
=> ( rp_is @ X2 @ X3 ) ) ) ) ) ) ) ).
thf(typ_diffprop1,type,
diffprop1: $i > $i > $i > $o ).
thf(def_diffprop1,definition,
( diffprop1
= ( ^ [X0: $i,X1: $i,X2: $i] :
( d_and @ ( rt_more @ X1 @ X2 )
@ ( ( rt_more @ X1 @ X2 )
=> ( rt_is @ X0 @ ( rt_mn @ X1 @ X2 ) ) ) ) ) ) ).
thf(typ_diffprop2,type,
diffprop2: $i > $i > $i > $i > $i > $o ).
thf(def_diffprop2,definition,
( diffprop2
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i] : ( and3 @ ( lrt @ X0 @ X3 ) @ ( urt @ X1 @ X4 ) @ ( diffprop1 @ X2 @ X3 @ X4 ) ) ) ) ).
thf(typ_rp_diffprop,type,
rp_diffprop: $i > $i > $i > $o ).
thf(def_rp_diffprop,definition,
( rp_diffprop
= ( ^ [X0: $i,X1: $i,X2: $i] :
( rt_some
@ ^ [X3: $i] : ( rt_some @ ( diffprop2 @ X0 @ X1 @ X2 @ X3 ) ) ) ) ) ).
thf(typ_diff,type,
diff: $i > $i > $i ).
thf(def_diff,definition,
( diff
= ( ^ [X0: $i,X1: $i] : ( d_Sep @ rat @ ( rp_diffprop @ X0 @ X1 ) ) ) ) ).
thf(satz140h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( cutprop @ ( diff @ X0 @ X1 ) ) ) ) ) ).
thf(typ_chi,type,
chi: $i > $i > $i ).
thf(def_chi,definition,
( chi
= ( ^ [X0: $i,X1: $i] : ( cutof @ ( diff @ X0 @ X1 ) ) ) ) ).
thf(satz140a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_some
@ ^ [X2: $i] : ( rp_is @ ( rp_pl @ X1 @ X2 ) @ X0 ) ) ) ) ) ).
thf(satz140,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_one
@ ^ [X2: $i] : ( rp_is @ ( rp_pl @ X1 @ X2 ) @ X0 ) ) ) ) ) ).
thf(typ_rp_mn,type,
rp_mn: $i > $i > $i ).
thf(def_rp_mn,definition,
( rp_mn
= ( ^ [X0: $i,X1: $i] :
( ind @ cut
@ ^ [X2: $i] : ( rp_is @ ( rp_pl @ X1 @ X2 ) @ X0 ) ) ) ) ).
thf(satz140c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_is @ ( rp_pl @ X1 @ ( rp_mn @ X0 @ X1 ) ) @ X0 ) ) ) ) ).
thf(satz140d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_is @ X0 @ ( rp_pl @ X1 @ ( rp_mn @ X0 @ X1 ) ) ) ) ) ) ).
thf(satz140e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_is @ ( rp_pl @ ( rp_mn @ X0 @ X1 ) @ X1 ) @ X0 ) ) ) ) ).
thf(satz140f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_is @ X0 @ ( rp_pl @ ( rp_mn @ X0 @ X1 ) @ X1 ) ) ) ) ) ).
thf(satz140g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ X0 @ X1 )
=> ( ( rp_is @ ( rp_pl @ X1 @ X2 ) @ X0 )
=> ( rp_is @ X2 @ ( rp_mn @ X0 @ X1 ) ) ) ) ) ) ) ).
thf(typ_prodprop1,type,
prodprop1: $i > $i > $i > $i > $i > $o ).
thf(def_prodprop1,definition,
( prodprop1
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i] : ( and3 @ ( lrt @ X0 @ X3 ) @ ( lrt @ X1 @ X4 ) @ ( rt_is @ X2 @ ( rt_ts @ X3 @ X4 ) ) ) ) ) ).
thf(typ_prodprop,type,
prodprop: $i > $i > $i > $o ).
thf(def_prodprop,definition,
( prodprop
= ( ^ [X0: $i,X1: $i,X2: $i] :
( rt_some
@ ^ [X3: $i] : ( rt_some @ ( prodprop1 @ X0 @ X1 @ X2 @ X3 ) ) ) ) ) ).
thf(typ_prod,type,
prod: $i > $i > $i ).
thf(def_prod,definition,
( prod
= ( ^ [X0: $i,X1: $i] : ( d_Sep @ rat @ ( prodprop @ X0 @ X1 ) ) ) ) ).
thf(satz141a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( urt @ X0 @ X2 )
=> ( all_of
@ ^ [X3: $i] : ( in @ X3 @ rat )
@ ^ [X3: $i] :
( ( urt @ X1 @ X3 )
=> ( d_not @ ( rt_in @ ( rt_ts @ X2 @ X3 ) @ ( prod @ X0 @ X1 ) ) ) ) ) ) ) ) ) ).
thf(typ_d_4141_v0,type,
d_4141_v0: $i > $i > $i ).
thf(def_d_4141_v0,definition,
( d_4141_v0
= ( ^ [X0: $i,X1: $i] : ( rt_ts @ ( rt_ov @ d_1rt @ X1 ) @ X0 ) ) ) ).
thf(satz141b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] : ( rt_is @ ( rt_ts @ ( rt_ov @ d_1rt @ X1 ) @ X0 ) @ ( rt_ov @ X0 @ X1 ) ) ) ) ).
thf(satz141c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] : ( rt_is @ ( rt_ov @ X0 @ X1 ) @ ( rt_ts @ ( rt_ov @ d_1rt @ X1 ) @ X0 ) ) ) ) ).
thf(satz141,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( cutprop @ ( prod @ X0 @ X1 ) ) ) ) ).
thf(typ_rp_ts,type,
rp_ts: $i > $i > $i ).
thf(def_rp_ts,definition,
( rp_ts
= ( ^ [X0: $i,X1: $i] : ( cutof @ ( prod @ X0 @ X1 ) ) ) ) ).
thf(satz142,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_is @ ( rp_ts @ X0 @ X1 ) @ ( rp_ts @ X1 @ X0 ) ) ) ) ).
thf(satz143,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] : ( rp_is @ ( rp_ts @ ( rp_ts @ X0 @ X1 ) @ X2 ) @ ( rp_ts @ X0 @ ( rp_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(typ_d_4144_x2,type,
d_4144_x2: $i > $i > $i ).
thf(def_d_4144_x2,definition,
( d_4144_x2
= ( ^ [X0: $i,X1: $i] : ( ite @ ( rt_moreis @ X0 @ X1 ) @ rat @ X0 @ X1 ) ) ) ).
thf(satz144,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] : ( rp_is @ ( rp_ts @ X0 @ ( rp_pl @ X1 @ X2 ) ) @ ( rp_pl @ ( rp_ts @ X0 @ X1 ) @ ( rp_ts @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz145a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_more @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz145b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ X0 @ X1 )
=> ( rp_is @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz145c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ X0 @ X1 )
=> ( rp_less @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz145d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ X0 @ X1 )
=> ( rp_more @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz145e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ X0 @ X1 )
=> ( rp_is @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz145f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ X0 @ X1 )
=> ( rp_less @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz145g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz145h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz145j,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz145k,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz146a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X2 ) )
=> ( rp_more @ X0 @ X1 ) ) ) ) ) ).
thf(satz146b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X2 ) )
=> ( rp_is @ X0 @ X1 ) ) ) ) ) ).
thf(satz146c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X2 ) )
=> ( rp_less @ X0 @ X1 ) ) ) ) ) ).
thf(satz146d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_more @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X2 @ X1 ) )
=> ( rp_more @ X0 @ X1 ) ) ) ) ) ).
thf(satz146e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X2 @ X1 ) )
=> ( rp_is @ X0 @ X1 ) ) ) ) ) ).
thf(satz146f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_less @ ( rp_ts @ X2 @ X0 ) @ ( rp_ts @ X2 @ X1 ) )
=> ( rp_less @ X0 @ X1 ) ) ) ) ) ).
thf(satz147,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_more @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz147a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_less @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz148a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_moreis @ X0 @ X1 )
=> ( ( rp_more @ X2 @ X3 )
=> ( rp_more @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz148b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_more @ X0 @ X1 )
=> ( ( rp_moreis @ X2 @ X3 )
=> ( rp_more @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz148c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( ( rp_less @ X2 @ X3 )
=> ( rp_less @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz148d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_less @ X0 @ X1 )
=> ( ( rp_lessis @ X2 @ X3 )
=> ( rp_less @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz149,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_moreis @ X0 @ X1 )
=> ( ( rp_moreis @ X2 @ X3 )
=> ( rp_moreis @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz149a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_lessis @ X0 @ X1 )
=> ( ( rp_lessis @ X2 @ X3 )
=> ( rp_lessis @ ( rp_ts @ X0 @ X2 ) @ ( rp_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(typ_ratset,type,
ratset: $i > $i ).
thf(def_ratset,definition,
( ratset
= ( ^ [X0: $i] :
( d_Sep @ rat
@ ^ [X1: $i] : ( rt_less @ X1 @ X0 ) ) ) ) ).
thf(satz150,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] : ( cutprop @ ( ratset @ X0 ) ) ) ).
thf(typ_rpofrt,type,
rpofrt: $i > $i ).
thf(def_rpofrt,definition,
( rpofrt
= ( ^ [X0: $i] : ( cutof @ ( ratset @ X0 ) ) ) ) ).
thf(typ_d_1rp,type,
d_1rp: $i ).
thf(def_d_1rp,definition,
( d_1rp
= ( rpofrt @ d_1rt ) ) ).
thf(satz151,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] : ( rp_is @ ( rp_ts @ X0 @ d_1rp ) @ X0 ) ) ).
thf(satz151a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] : ( rp_is @ X0 @ ( rp_ts @ X0 @ d_1rp ) ) ) ).
thf(satz151b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] : ( rp_is @ ( rp_ts @ d_1rp @ X0 ) @ X0 ) ) ).
thf(satz151c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] : ( rp_is @ X0 @ ( rp_ts @ d_1rp @ X0 ) ) ) ).
thf(typ_invprop1,type,
invprop1: $i > $i > $i > $o ).
thf(def_invprop1,definition,
( invprop1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( d_and @ ( urt @ X0 @ X2 ) @ ( rt_less @ X2 @ X1 ) ) ) ) ).
thf(typ_invprop2,type,
invprop2: $i > $i > $i > $o ).
thf(def_invprop2,definition,
( invprop2
= ( ^ [X0: $i,X1: $i,X2: $i] : ( and3 @ ( urt @ X0 @ X2 ) @ ( rt_some @ ( invprop1 @ X0 @ X2 ) ) @ ( rt_is @ X1 @ ( rt_ov @ d_1rt @ X2 ) ) ) ) ) ).
thf(typ_invprop,type,
invprop: $i > $i > $o ).
thf(def_invprop,definition,
( invprop
= ( ^ [X0: $i,X1: $i] : ( rt_some @ ( invprop2 @ X0 @ X1 ) ) ) ) ).
thf(typ_inv,type,
inv: $i > $i ).
thf(def_inv,definition,
( inv
= ( ^ [X0: $i] : ( d_Sep @ rat @ ( invprop @ X0 ) ) ) ) ).
thf(typ_d_2x0,type,
d_2x0: $i > $i ).
thf(def_d_2x0,definition,
( d_2x0
= ( ^ [X0: $i] : ( rt_pl @ X0 @ X0 ) ) ) ).
thf(typ_d_4152_chi,type,
d_4152_chi: $i > $i ).
thf(def_d_4152_chi,definition,
( d_4152_chi
= ( ^ [X0: $i] : ( cutof @ ( inv @ X0 ) ) ) ) ).
thf(satz152,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( rp_some
@ ^ [X1: $i] : ( rp_is @ ( rp_ts @ X0 @ X1 ) @ d_1rp ) ) ) ).
thf(satz153b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ cut )
@ ^ [X3: $i] :
( ( rp_is @ ( rp_ts @ X1 @ X2 ) @ X0 )
=> ( ( rp_is @ ( rp_ts @ X1 @ X3 ) @ X0 )
=> ( rp_is @ X2 @ X3 ) ) ) ) ) ) ) ).
thf(typ_d_4153_chi,type,
d_4153_chi: $i > $i > $i ).
thf(def_d_4153_chi,definition,
( d_4153_chi
= ( ^ [X0: $i,X1: $i] : ( rp_ts @ X1 @ X0 ) ) ) ).
thf(satz153a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( rp_some
@ ^ [X2: $i] : ( rp_is @ ( rp_ts @ X1 @ X2 ) @ X0 ) ) ) ) ).
thf(satz153,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( rp_one
@ ^ [X2: $i] : ( rp_is @ ( rp_ts @ X1 @ X2 ) @ X0 ) ) ) ) ).
thf(typ_rp_ov,type,
rp_ov: $i > $i > $i ).
thf(def_rp_ov,definition,
( rp_ov
= ( ^ [X0: $i,X1: $i] :
( ind @ cut
@ ^ [X2: $i] : ( rp_is @ ( rp_ts @ X1 @ X2 ) @ X0 ) ) ) ) ).
thf(satz153c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_is @ ( rp_ts @ X1 @ ( rp_ov @ X0 @ X1 ) ) @ X0 ) ) ) ).
thf(satz153d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_is @ X0 @ ( rp_ts @ X1 @ ( rp_ov @ X0 @ X1 ) ) ) ) ) ).
thf(satz153e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_is @ ( rp_ts @ ( rp_ov @ X0 @ X1 ) @ X1 ) @ X0 ) ) ) ).
thf(satz153f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] : ( rp_is @ X0 @ ( rp_ts @ ( rp_ov @ X0 @ X1 ) @ X1 ) ) ) ) ).
thf(satz153g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ cut )
@ ^ [X2: $i] :
( ( rp_is @ ( rp_ts @ X1 @ X2 ) @ X0 )
=> ( rp_is @ X2 @ ( rp_ov @ X0 @ X1 ) ) ) ) ) ) ).
thf(typ_ratrp,type,
ratrp: $i > $o ).
thf(def_ratrp,definition,
( ratrp
= ( image @ rat @ cut @ ( d_Sigma @ rat @ rpofrt ) ) ) ).
thf(typ_rpofnt,type,
rpofnt: $i > $i ).
thf(def_rpofnt,definition,
( rpofnt
= ( ^ [X0: $i] : ( rpofrt @ ( rtofn @ X0 ) ) ) ) ).
thf(typ_natrp,type,
natrp: $i > $o ).
thf(def_natrp,definition,
( natrp
= ( image @ nat @ cut @ ( d_Sigma @ nat @ rpofnt ) ) ) ).
thf(satz154a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rt_more @ X0 @ X1 )
=> ( rp_more @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(satz154b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rt_is @ X0 @ X1 )
=> ( rp_is @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(satz154c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rt_less @ X0 @ X1 )
=> ( rp_less @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(satz154d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rp_more @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) )
=> ( rt_more @ X0 @ X1 ) ) ) ) ).
thf(satz154e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rp_is @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) )
=> ( rt_is @ X0 @ X1 ) ) ) ) ).
thf(satz154f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rp_less @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) )
=> ( rt_less @ X0 @ X1 ) ) ) ) ).
thf(typ_rtofrp,type,
rtofrp: $i > $i ).
thf(def_rtofrp,definition,
( rtofrp
= ( soft @ rat @ cut @ ( d_Sigma @ rat @ rpofrt ) ) ) ).
thf(typ_ntofrp,type,
ntofrp: $i > $i ).
thf(def_ntofrp,definition,
( ntofrp
= ( soft @ nat @ cut @ ( d_Sigma @ nat @ rpofnt ) ) ) ).
thf(typ_u01,type,
u01: $i > $i > $i > $i ).
thf(def_u01,definition,
( u01
= ( ^ [X0: $i,X1: $i,X2: $i] : ( rt_ov @ X2 @ ( rt_pl @ X0 @ X1 ) ) ) ) ).
thf(satz155a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] : ( rp_is @ ( rpofrt @ ( rt_pl @ X0 @ X1 ) ) @ ( rp_pl @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(satz155b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rt_more @ X0 @ X1 )
=> ( rp_is @ ( rpofrt @ ( rt_mn @ X0 @ X1 ) ) @ ( rp_mn @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) ) ) ) ) ) ).
thf(satz155c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] : ( rp_is @ ( rpofrt @ ( rt_ts @ X0 @ X1 ) ) @ ( rp_ts @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(satz155d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ rat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] : ( rp_is @ ( rpofrt @ ( rt_ov @ X0 @ X1 ) ) @ ( rp_ov @ ( rpofrt @ X0 ) @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(satz155e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( rp_is @ ( rpofnt @ ( n_pl @ X0 @ X1 ) ) @ ( rp_pl @ ( rpofnt @ X0 ) @ ( rpofnt @ X1 ) ) ) ) ) ).
thf(satz155f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( rp_is @ ( rpofnt @ ( n_ts @ X0 @ X1 ) ) @ ( rp_ts @ ( rpofnt @ X0 ) @ ( rpofnt @ X1 ) ) ) ) ) ).
thf(typ_nt_natt,type,
nt_natt: $i ).
thf(def_nt_natt,definition,
( nt_natt
= ( d_Sep @ cut @ natrp ) ) ).
thf(typ_nttofrp,type,
nttofrp: $i > $i ).
thf(def_nttofrp,definition,
( nttofrp
= ( out @ cut @ natrp ) ) ).
thf(typ_rp_nt_is,type,
rp_nt_is: $i > $i > $o ).
thf(def_rp_nt_is,definition,
( rp_nt_is
= ( e_is @ nt_natt ) ) ).
thf(typ_rp_nt_nis,type,
rp_nt_nis: $i > $i > $o ).
thf(def_rp_nt_nis,definition,
( rp_nt_nis
= ( ^ [X0: $i,X1: $i] : ( d_not @ ( rp_nt_is @ X0 @ X1 ) ) ) ) ).
thf(typ_rp_nt_all,type,
rp_nt_all: ( $i > $o ) > $o ).
thf(def_rp_nt_all,definition,
( rp_nt_all
= ( all @ nt_natt ) ) ).
thf(typ_rp_nt_some,type,
rp_nt_some: ( $i > $o ) > $o ).
thf(def_rp_nt_some,definition,
( rp_nt_some
= ( l_some @ nt_natt ) ) ).
thf(typ_rp_nt_one,type,
rp_nt_one: ( $i > $o ) > $o ).
thf(def_rp_nt_one,definition,
( rp_nt_one
= ( one @ nt_natt ) ) ).
thf(typ_rp_nt_in,type,
rp_nt_in: $i > $i > $o ).
thf(def_rp_nt_in,definition,
( rp_nt_in
= ( esti @ nt_natt ) ) ).
thf(typ_rpofntt,type,
rpofntt: $i > $i ).
thf(def_rpofntt,definition,
( rpofntt
= ( e_in @ cut @ natrp ) ) ).
thf(typ_nttofnt,type,
nttofnt: $i > $i ).
thf(def_nttofnt,definition,
( nttofnt
= ( ^ [X0: $i] : ( nttofrp @ ( rpofnt @ X0 ) ) ) ) ).
thf(typ_ntofntt,type,
ntofntt: $i > $i ).
thf(def_ntofntt,definition,
( ntofntt
= ( ^ [X0: $i] : ( ntofrp @ ( rpofntt @ X0 ) ) ) ) ).
thf(typ_rp_nt_1t,type,
rp_nt_1t: $i ).
thf(def_rp_nt_1t,definition,
( rp_nt_1t
= ( nttofnt @ n_1 ) ) ).
thf(typ_nt_suct,type,
nt_suct: $i ).
thf(def_nt_suct,definition,
( nt_suct
= ( d_Sigma @ nt_natt
@ ^ [X0: $i] : ( nttofnt @ ( ordsucc @ ( ntofntt @ X0 ) ) ) ) ) ).
thf(satz156a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nt_natt )
@ ^ [X0: $i] : ( rp_nt_nis @ ( ap @ nt_suct @ X0 ) @ rp_nt_1t ) ) ).
thf(satz156b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nt_natt )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nt_natt )
@ ^ [X1: $i] :
( ( rp_nt_is @ ( ap @ nt_suct @ X0 ) @ ( ap @ nt_suct @ X1 ) )
=> ( rp_nt_is @ X0 @ X1 ) ) ) ) ).
thf(typ_rp_nt_cond1,type,
rp_nt_cond1: $i > $o ).
thf(def_rp_nt_cond1,definition,
( rp_nt_cond1
= ( rp_nt_in @ rp_nt_1t ) ) ).
thf(typ_rp_nt_cond2,type,
rp_nt_cond2: $i > $o ).
thf(def_rp_nt_cond2,definition,
( rp_nt_cond2
= ( ^ [X0: $i] :
( rp_nt_all
@ ^ [X1: $i] : ( imp @ ( rp_nt_in @ X1 @ X0 ) @ ( rp_nt_in @ ( ap @ nt_suct @ X1 ) @ X0 ) ) ) ) ) ).
thf(typ_d_5156_prop1,type,
d_5156_prop1: $i > $i > $o ).
thf(def_d_5156_prop1,definition,
( d_5156_prop1
= ( ^ [X0: $i,X1: $i] : ( rp_nt_in @ ( nttofnt @ X1 ) @ X0 ) ) ) ).
thf(satz156c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ ( power @ nt_natt ) )
@ ^ [X0: $i] :
( ( rp_nt_cond1 @ X0 )
=> ( ( rp_nt_cond2 @ X0 )
=> ( all_of
@ ^ [X1: $i] : ( in @ X1 @ nt_natt )
@ ^ [X1: $i] : ( rp_nt_in @ X1 @ X0 ) ) ) ) ) ).
thf(typ_ratt,type,
ratt: $i ).
thf(def_ratt,definition,
( ratt
= ( d_Sep @ cut @ ratrp ) ) ).
thf(typ_rttofrp,type,
rttofrp: $i > $i ).
thf(def_rttofrp,definition,
( rttofrp
= ( out @ cut @ ratrp ) ) ).
thf(typ_rtt_is,type,
rtt_is: $i > $i > $o ).
thf(def_rtt_is,definition,
( rtt_is
= ( e_is @ ratt ) ) ).
thf(typ_rtt_nis,type,
rtt_nis: $i > $i > $o ).
thf(def_rtt_nis,definition,
( rtt_nis
= ( ^ [X0: $i,X1: $i] : ( d_not @ ( rtt_is @ X0 @ X1 ) ) ) ) ).
thf(typ_rtt_all,type,
rtt_all: ( $i > $o ) > $o ).
thf(def_rtt_all,definition,
( rtt_all
= ( all @ ratt ) ) ).
thf(typ_rtt_some,type,
rtt_some: ( $i > $o ) > $o ).
thf(def_rtt_some,definition,
( rtt_some
= ( l_some @ ratt ) ) ).
thf(typ_rtt_one,type,
rtt_one: ( $i > $o ) > $o ).
thf(def_rtt_one,definition,
( rtt_one
= ( one @ ratt ) ) ).
thf(typ_rpofrtt,type,
rpofrtt: $i > $i ).
thf(def_rpofrtt,definition,
( rpofrtt
= ( e_in @ cut @ ratrp ) ) ).
thf(typ_rttofrt,type,
rttofrt: $i > $i ).
thf(def_rttofrt,definition,
( rttofrt
= ( ^ [X0: $i] : ( rttofrp @ ( rpofrt @ X0 ) ) ) ) ).
thf(typ_rtofrtt,type,
rtofrtt: $i > $i ).
thf(def_rtofrtt,definition,
( rtofrtt
= ( ^ [X0: $i] : ( rtofrp @ ( rpofrtt @ X0 ) ) ) ) ).
thf(typ_d_5157_s1,type,
d_5157_s1: $i > $i ).
thf(def_d_5157_s1,definition,
( d_5157_s1
= ( ^ [X0: $i] : ( d_Sep @ rat @ ( urt @ X0 ) ) ) ) ).
thf(satz157a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( ( ratrp @ X0 )
=> ( rt_min @ ( d_Sep @ rat @ ( urt @ X0 ) ) @ ( rtofrp @ X0 ) ) ) ) ).
thf(satz157b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( ( ratrp @ X0 )
=> ( rt_some @ ( rt_min @ ( d_Sep @ rat @ ( urt @ X0 ) ) ) ) ) ) ).
thf(satz157c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rt_min @ ( d_Sep @ rat @ ( urt @ X0 ) ) @ X1 )
=> ( rp_is @ X0 @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(satz157d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( ( rt_some @ ( rt_min @ ( d_Sep @ rat @ ( urt @ X0 ) ) ) )
=> ( ratrp @ X0 ) ) ) ).
thf(satz158a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( lrt @ X0 @ X1 )
=> ( rp_less @ ( rpofrt @ X1 ) @ X0 ) ) ) ) ).
thf(satz158b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( urt @ X0 @ X1 )
=> ( rp_moreis @ ( rpofrt @ X1 ) @ X0 ) ) ) ) ).
thf(satz158c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rp_less @ ( rpofrt @ X1 ) @ X0 )
=> ( lrt @ X0 @ X1 ) ) ) ) ).
thf(satz158d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ rat )
@ ^ [X1: $i] :
( ( rp_moreis @ ( rpofrt @ X1 ) @ X0 )
=> ( urt @ X0 @ X1 ) ) ) ) ).
thf(satz159,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_less @ X0 @ X1 )
=> ( rt_some
@ ^ [X2: $i] : ( d_and @ ( rp_less @ X0 @ ( rpofrt @ X2 ) ) @ ( rp_less @ ( rpofrt @ X2 ) @ X1 ) ) ) ) ) ) ).
thf(satz159a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_less @ X0 @ X1 )
=> ( rp_some
@ ^ [X2: $i] : ( and3 @ ( ratrp @ X2 ) @ ( rp_less @ X0 @ X2 ) @ ( rp_less @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz159app,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( ( rp_less @ X0 @ X1 )
=> ! [X2: $o] :
( ( all_of
@ ^ [X3: $i] : ( in @ X3 @ rat )
@ ^ [X3: $i] :
( ( rp_less @ X0 @ ( rpofrt @ X3 ) )
=> ( ( rp_less @ ( rpofrt @ X3 ) @ X1 )
=> X2 ) ) )
=> X2 ) ) ) ) ).
thf(typ_d_5160_nm,type,
d_5160_nm: $i > $i > $i > $i ).
thf(def_d_5160_nm,definition,
( d_5160_nm
= ( ^ [X0: $i,X1: $i,X2: $i] : ( rp_mn @ ( rpofrt @ X2 ) @ ( rp_ts @ X0 @ X1 ) ) ) ) ).
thf(typ_d_5160_dn,type,
d_5160_dn: $i > $i > $i ).
thf(def_d_5160_dn,definition,
( d_5160_dn
= ( ^ [X0: $i,X1: $i] : ( rp_pl @ ( rp_pl @ X0 @ X1 ) @ d_1rp ) ) ) ).
thf(typ_d_5160_fr,type,
d_5160_fr: $i > $i > $i > $i ).
thf(def_d_5160_fr,definition,
( d_5160_fr
= ( ^ [X0: $i,X1: $i,X2: $i] : ( rp_ov @ ( d_5160_nm @ X0 @ X1 @ X2 ) @ ( d_5160_dn @ X0 @ X1 ) ) ) ) ).
thf(typ_zeta,type,
zeta: $i > $i > $i > $i ).
thf(def_zeta,definition,
( zeta
= ( ^ [X0: $i,X1: $i,X2: $i] : ( ite @ ( rp_less @ ( d_5160_fr @ X0 @ X1 @ X2 ) @ d_1rp ) @ cut @ ( d_5160_fr @ X0 @ X1 @ X2 ) @ d_1rp ) ) ) ).
thf(typ_d_5160_xr,type,
d_5160_xr: $i > $i > $i ).
thf(def_d_5160_xr,definition,
( d_5160_xr
= ( ^ [X0: $i,X1: $i] : ( rpofrt @ ( rt_ov @ X0 @ X1 ) ) ) ) ).
thf(typ_d_5160_y0,type,
d_5160_y0: $i > $i ).
thf(def_d_5160_y0,definition,
( d_5160_y0
= ( ^ [X0: $i] : X0 ) ) ).
thf(typ_d_5160_yr,type,
d_5160_yr: $i > $i ).
thf(def_d_5160_yr,definition,
( d_5160_yr
= ( ^ [X0: $i] : ( rpofrt @ ( d_5160_y0 @ X0 ) ) ) ) ).
thf(typ_d_5160_prop1,type,
d_5160_prop1: $i > $i > $i > $i > $i > $o ).
thf(def_d_5160_prop1,definition,
( d_5160_prop1
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i] : ( and3 @ ( rp_more @ ( rpofrt @ X3 ) @ X0 ) @ ( rp_more @ ( rpofrt @ X4 ) @ X1 ) @ ( rt_is @ ( rt_ts @ X3 @ X4 ) @ X2 ) ) ) ) ).
thf(typ_d_5160_prop2,type,
d_5160_prop2: $i > $i > $i > $o ).
thf(def_d_5160_prop2,definition,
( d_5160_prop2
= ( ^ [X0: $i,X1: $i,X2: $i] :
( rt_some
@ ^ [X3: $i] : ( rt_some @ ( d_5160_prop1 @ X0 @ X1 @ X2 @ X3 ) ) ) ) ) ).
thf(satz160,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( rp_more @ ( rpofrt @ X2 ) @ ( rp_ts @ X0 @ X1 ) )
=> ( rt_some
@ ^ [X3: $i] :
( rt_some
@ ^ [X4: $i] : ( and3 @ ( rp_more @ ( rpofrt @ X3 ) @ X0 ) @ ( rp_more @ ( rpofrt @ X4 ) @ X1 ) @ ( rt_is @ ( rt_ts @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) ) ).
thf(satz160app,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ cut )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ rat )
@ ^ [X2: $i] :
( ( rp_more @ ( rpofrt @ X2 ) @ ( rp_ts @ X0 @ X1 ) )
=> ! [X3: $o] :
( ( all_of
@ ^ [X4: $i] : ( in @ X4 @ rat )
@ ^ [X4: $i] :
( ( rp_more @ ( rpofrt @ X4 ) @ X0 )
=> ( all_of
@ ^ [X5: $i] : ( in @ X5 @ rat )
@ ^ [X5: $i] :
( ( rp_more @ ( rpofrt @ X5 ) @ X1 )
=> ( ( rt_is @ ( rt_ts @ X4 @ X5 ) @ X2 )
=> X3 ) ) ) ) )
=> X3 ) ) ) ) ) ).
thf(typ_d_5161_min,type,
d_5161_min: $i > $i > $i ).
thf(def_d_5161_min,definition,
( d_5161_min
= ( ^ [X0: $i,X1: $i] : ( ite @ ( rp_less @ X0 @ X1 ) @ cut @ X0 @ X1 ) ) ) ).
thf(typ_d_5161_max,type,
d_5161_max: $i > $i > $i ).
thf(def_d_5161_max,definition,
( d_5161_max
= ( ^ [X0: $i,X1: $i] : ( ite @ ( rp_more @ X0 @ X1 ) @ cut @ X0 @ X1 ) ) ) ).
thf(typ_sq1,type,
sq1: $i > $i ).
thf(def_sq1,definition,
( sq1
= ( ^ [X0: $i] : ( rp_ts @ X0 @ X0 ) ) ) ).
thf(typ_sqrtset,type,
sqrtset: $i > $i ).
thf(def_sqrtset,definition,
( sqrtset
= ( ^ [X0: $i] :
( d_Sep @ rat
@ ^ [X1: $i] : ( rp_less @ ( rp_ts @ ( rpofrt @ X1 ) @ ( rpofrt @ X1 ) ) @ X0 ) ) ) ) ).
thf(typ_d_5161_nm,type,
d_5161_nm: $i > $i > $i ).
thf(def_d_5161_nm,definition,
( d_5161_nm
= ( ^ [X0: $i,X1: $i] : ( rp_mn @ X0 @ ( rp_ts @ ( rpofrt @ X1 ) @ ( rpofrt @ X1 ) ) ) ) ) ).
thf(typ_d_5161_dn,type,
d_5161_dn: $i > $i ).
thf(def_d_5161_dn,definition,
( d_5161_dn
= ( ^ [X0: $i] : ( rp_pl @ ( rpofrt @ X0 ) @ ( rp_pl @ ( rpofrt @ X0 ) @ d_1rp ) ) ) ) ).
thf(typ_d_5161_fr,type,
d_5161_fr: $i > $i > $i ).
thf(def_d_5161_fr,definition,
( d_5161_fr
= ( ^ [X0: $i,X1: $i] : ( rp_ov @ ( d_5161_nm @ X0 @ X1 ) @ ( d_5161_dn @ X1 ) ) ) ) ).
thf(typ_rtc,type,
rtc: $i > $i ).
thf(def_rtc,definition,
( rtc
= ( ^ [X0: $i] : ( cutof @ ( sqrtset @ X0 ) ) ) ) ).
thf(typ_d_5161_xm,type,
d_5161_xm: $i > $i > $i ).
thf(def_d_5161_xm,definition,
( d_5161_xm
= ( ^ [X0: $i,X1: $i] : ( ite @ ( rt_more @ X0 @ X1 ) @ rat @ X0 @ X1 ) ) ) ).
thf(typ_xrm,type,
xrm: $i > $i > $i ).
thf(def_xrm,definition,
( xrm
= ( ^ [X0: $i,X1: $i] : ( rpofrt @ ( d_5161_xm @ X0 @ X1 ) ) ) ) ).
thf(typ_d_5161_ym,type,
d_5161_ym: $i > $i > $i ).
thf(def_d_5161_ym,definition,
( d_5161_ym
= ( ^ [X0: $i,X1: $i] : ( ite @ ( rt_less @ X0 @ X1 ) @ rat @ X0 @ X1 ) ) ) ).
thf(typ_yrm,type,
yrm: $i > $i > $i ).
thf(def_yrm,definition,
( yrm
= ( ^ [X0: $i,X1: $i] : ( rpofrt @ ( d_5161_ym @ X0 @ X1 ) ) ) ) ).
thf(satz161,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ cut )
@ ^ [X0: $i] :
( rp_one
@ ^ [X1: $i] : ( rp_is @ ( rp_ts @ X1 @ X1 ) @ X0 ) ) ) ).
thf(typ_irratrp,type,
irratrp: $i > $o ).
thf(def_irratrp,definition,
( irratrp
= ( ^ [X0: $i] : ( d_not @ ( ratrp @ X0 ) ) ) ) ).
thf(typ_d_5162_prop1,type,
d_5162_prop1: $i > $i > $o ).
thf(def_d_5162_prop1,definition,
( d_5162_prop1
= ( ^ [X0: $i,X1: $i] : ( n_eq @ ( n_tf @ ( n_fr @ X1 @ X0 ) @ ( n_fr @ X1 @ X0 ) ) @ ( n_fr @ ( ordsucc @ n_1 ) @ n_1 ) ) ) ) ).
thf(typ_d_5162_prop2,type,
d_5162_prop2: $i > $o ).
thf(def_d_5162_prop2,definition,
( d_5162_prop2
= ( ^ [X0: $i] : ( n_some @ ( d_5162_prop1 @ X0 ) ) ) ) ).
thf(typ_d_5162_prop3,type,
d_5162_prop3: $o ).
thf(def_d_5162_prop3,definition,
( d_5162_prop3
= ( n_some @ d_5162_prop2 ) ) ).
thf(typ_d_5162_y,type,
d_5162_y: $i ).
thf(def_d_5162_y,definition,
( d_5162_y
= ( ind @ nat @ ( min @ d_5162_prop2 ) ) ) ).
thf(typ_ksi,type,
ksi: $i ).
thf(def_ksi,definition,
( ksi
= ( ind @ cut
@ ^ [X0: $i] : ( rp_is @ ( rp_ts @ X0 @ X0 ) @ ( rpofnt @ ( ordsucc @ n_1 ) ) ) ) ) ).
thf(typ_d_5162_x0,type,
d_5162_x0: $i ).
thf(def_d_5162_x0,definition,
( d_5162_x0
= ( rtofrp @ ksi ) ) ).
thf(satz162,axiom,
rp_some @ irratrp ).
thf(typ_sqrt,type,
sqrt: $i > $i ).
thf(def_sqrt,definition,
( sqrt
= ( ^ [X0: $i] :
( ind @ cut
@ ^ [X1: $i] : ( rp_is @ ( rp_ts @ X1 @ X1 ) @ X0 ) ) ) ) ).
thf(typ_iiia_x0,type,
iiia_x0: $i > $i ).
thf(def_iiia_x0,definition,
( iiia_x0
= ( ^ [X0: $i] : ( rtofn @ ( ntofrp @ X0 ) ) ) ) ).
thf(typ_xpy,type,
xpy: $i > $i > $i ).
thf(def_xpy,definition,
( xpy
= ( ^ [X0: $i,X1: $i] : ( nofrt @ ( rt_pl @ ( iiia_x0 @ X0 ) @ ( iiia_x0 @ X1 ) ) ) ) ) ).
thf(typ_xty,type,
xty: $i > $i > $i ).
thf(def_xty,definition,
( xty
= ( ^ [X0: $i,X1: $i] : ( nofrt @ ( rt_ts @ ( iiia_x0 @ X0 ) @ ( iiia_x0 @ X1 ) ) ) ) ) ).
thf(typ_xmy,type,
xmy: $i > $i > $i ).
thf(def_xmy,definition,
( xmy
= ( ^ [X0: $i,X1: $i] : ( nofrt @ ( rt_mn @ ( iiia_x0 @ X0 ) @ ( iiia_x0 @ X1 ) ) ) ) ) ).
thf(typ_dif,type,
dif: $i ).
thf(def_dif,definition,
( dif
= ( pair1type @ cut ) ) ).
thf(typ_rp_df,type,
rp_df: $i > $i > $i ).
thf(def_rp_df,definition,
( rp_df
= ( pair1 @ cut ) ) ).
thf(typ_stm,type,
stm: $i > $i ).
thf(def_stm,definition,
( stm
= ( first1 @ cut ) ) ).
thf(typ_std,type,
std: $i > $i ).
thf(def_std,definition,
( std
= ( second1 @ cut ) ) ).
%------------------------------------------------------------------------------