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      :

% 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 ) )).

%------------------------------------------------------------------------------