## TPTP Axioms File: NUM007^3.ax

```%------------------------------------------------------------------------------
% File     : NUM007^3 : TPTP v7.5.0. 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 (   0 unit; 117 type; 117 defn)
%            Number of atoms       : 3722 ( 117 equality;1404 variable)
%            Maximal formula depth :   21 (   8 average)
%            Number of connectives : 3227 (   0   ~;   0   |;   0   &;3065   @)
%                                         (   0 <=>; 162  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  209 ( 209   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  177 ( 117   :;   0   =)
%            Number of variables   :  953 (   1 sgn;   3   !;   0   ?; 950   ^)
%                                         ( 953   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      :

%------------------------------------------------------------------------------
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 ) )).

%------------------------------------------------------------------------------
```