TPTP Axioms File: NUM007^2.ax


%------------------------------------------------------------------------------
% File     : NUM007^2 : TPTP v7.5.0. Released v7.1.0.
% Domain   : Number Theory
% Axioms   : Grundlagen chapter 2
% Version  : [Bro17] axioms : Especial.
% English  :

% Refs     : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source   : [Bro17]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :  278 (   0 unit;  81 type;  81 defn)
%            Number of atoms       : 2894 (  81 equality;1066 variable)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives : 2535 (   0   ~;   0   |;   0   &;2417   @)
%                                         (   0 <=>; 118  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  141 ( 141   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  133 (  81   :;   0   =)
%            Number of variables   :  723 (   1 sgn;   1   !;   0   ?; 722   ^)
%                                         ( 723   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      :

% Comments :
%------------------------------------------------------------------------------
thf(typ_inf,type,(
    inf: $i > $i > $o )).

thf(def_inf,definition,
    ( inf
    = ( esti @ frac ) )).

thf(typ_rat,type,(
    rat: $i )).

thf(def_rat,definition,
    ( rat
    = ( ect @ frac @ n_eq ) )).

thf(typ_rt_is,type,(
    rt_is: $i > $i > $o )).

thf(def_rt_is,definition,
    ( rt_is
    = ( e_is @ rat ) )).

thf(typ_rt_nis,type,(
    rt_nis: $i > $i > $o )).

thf(def_rt_nis,definition,
    ( rt_nis
    = ( ^ [X0: $i,X1: $i] :
          ( d_not @ ( rt_is @ X0 @ X1 ) ) ) )).

thf(typ_rt_some,type,(
    rt_some: ( $i > $o ) > $o )).

thf(def_rt_some,definition,
    ( rt_some
    = ( l_some @ rat ) )).

thf(typ_rt_all,type,(
    rt_all: ( $i > $o ) > $o )).

thf(def_rt_all,definition,
    ( rt_all
    = ( all @ rat ) )).

thf(typ_rt_one,type,(
    rt_one: ( $i > $o ) > $o )).

thf(def_rt_one,definition,
    ( rt_one
    = ( one @ rat ) )).

thf(typ_rt_in,type,(
    rt_in: $i > $i > $o )).

thf(def_rt_in,definition,
    ( rt_in
    = ( esti @ rat ) )).

thf(typ_ratof,type,(
    ratof: $i > $i )).

thf(def_ratof,definition,
    ( ratof
    = ( ectelt @ frac @ n_eq ) )).

thf(typ_class,type,(
    class: $i > $i )).

thf(def_class,definition,
    ( class
    = ( ecect @ frac @ n_eq ) )).

thf(typ_fixf,type,(
    fixf: $i > $i > $o )).

thf(def_fixf,definition,
    ( fixf
    = ( fixfu2 @ frac @ n_eq ) )).

thf(typ_indrat,type,(
    indrat: $i > $i > $i > $i > $i )).

thf(def_indrat,definition,
    ( indrat
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( indeq2 @ frac @ n_eq @ X2 @ X3 @ X0 @ X1 ) ) )).

thf(satz78,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( rt_is @ X0 @ X0 ) )).

thf(satz79,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_is @ X0 @ X1 )
           => ( rt_is @ X1 @ X0 ) ) ) )).

thf(satz80,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ X0 @ X1 )
               => ( ( rt_is @ X1 @ X2 )
                 => ( rt_is @ X0 @ X2 ) ) ) ) ) )).

thf(typ_rt_more,type,(
    rt_more: $i > $i > $o )).

thf(def_rt_more,definition,
    ( rt_more
    = ( ^ [X0: $i,X1: $i] :
          ( l_some @ frac
          @ ^ [X2: $i] :
              ( l_some @ frac
              @ ^ [X3: $i] :
                  ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( moref @ X2 @ X3 ) ) ) ) ) )).

thf(typ_propm,type,(
    propm: $i > $i > $i > $i > $o )).

thf(def_propm,definition,
    ( propm
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( moref @ X2 @ X3 ) ) ) )).

thf(typ_rt_less,type,(
    rt_less: $i > $i > $o )).

thf(def_rt_less,definition,
    ( rt_less
    = ( ^ [X0: $i,X1: $i] :
          ( l_some @ frac
          @ ^ [X2: $i] :
              ( l_some @ frac
              @ ^ [X3: $i] :
                  ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( lessf @ X2 @ X3 ) ) ) ) ) )).

thf(typ_propl,type,(
    propl: $i > $i > $i > $i > $o )).

thf(def_propl,definition,
    ( propl
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( lessf @ X2 @ X3 ) ) ) )).

thf(satz81,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( orec3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz81a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( or3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz81b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ec3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz82,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_less @ X1 @ X0 ) ) ) )).

thf(satz83,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_less @ X0 @ X1 )
           => ( rt_more @ X1 @ X0 ) ) ) )).

thf(typ_rt_moreis,type,(
    rt_moreis: $i > $i > $o )).

thf(def_rt_moreis,definition,
    ( rt_moreis
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( rt_more @ X0 @ X1 ) @ ( rt_is @ X0 @ X1 ) ) ) )).

thf(typ_rt_lessis,type,(
    rt_lessis: $i > $i > $o )).

thf(def_rt_lessis,definition,
    ( rt_lessis
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( rt_less @ X0 @ X1 ) @ ( rt_is @ X0 @ X1 ) ) ) )).

thf(satz81c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_moreis @ X0 @ X1 )
           => ( d_not @ ( rt_less @ X0 @ X1 ) ) ) ) )).

thf(satz81d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_lessis @ X0 @ X1 )
           => ( d_not @ ( rt_more @ X0 @ X1 ) ) ) ) )).

thf(satz81e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_more @ X0 @ X1 ) )
           => ( rt_lessis @ X0 @ X1 ) ) ) )).

thf(satz81f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_less @ X0 @ X1 ) )
           => ( rt_moreis @ X0 @ X1 ) ) ) )).

thf(satz81g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( d_not @ ( rt_lessis @ X0 @ X1 ) ) ) ) )).

thf(satz81h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_less @ X0 @ X1 )
           => ( d_not @ ( rt_moreis @ X0 @ X1 ) ) ) ) )).

thf(satz81j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_moreis @ X0 @ X1 ) )
           => ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz81k,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_lessis @ X0 @ X1 ) )
           => ( rt_more @ X0 @ X1 ) ) ) )).

thf(satz84,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_moreis @ X0 @ X1 )
           => ( rt_lessis @ X1 @ X0 ) ) ) )).

thf(satz85,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_lessis @ X0 @ X1 )
           => ( rt_moreis @ X1 @ X0 ) ) ) )).

thf(satz86,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( ( rt_less @ X1 @ X2 )
                 => ( rt_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz87a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_lessis @ X0 @ X1 )
               => ( ( rt_less @ X1 @ X2 )
                 => ( rt_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz87b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( ( rt_lessis @ X1 @ X2 )
                 => ( rt_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz87c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_moreis @ X0 @ X1 )
               => ( ( rt_more @ X1 @ X2 )
                 => ( rt_more @ X0 @ X2 ) ) ) ) ) )).

thf(satz87d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ X0 @ X1 )
               => ( ( rt_moreis @ X1 @ X2 )
                 => ( rt_more @ X0 @ X2 ) ) ) ) ) )).

thf(satz88,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_lessis @ X0 @ X1 )
               => ( ( rt_lessis @ X1 @ X2 )
                 => ( rt_lessis @ X0 @ X2 ) ) ) ) ) )).

thf(satz89,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( rt_some
        @ ^ [X1: $i] :
            ( rt_more @ X1 @ X0 ) ) )).

thf(satz90,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( rt_some
        @ ^ [X1: $i] :
            ( rt_less @ X1 @ X0 ) ) )).

thf(satz91,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_less @ X0 @ X1 )
           => ( rt_some
              @ ^ [X2: $i] :
                  ( d_and @ ( rt_less @ X0 @ X2 ) @ ( rt_less @ X2 @ X1 ) ) ) ) ) )).

thf(typ_plusfrt,type,(
    plusfrt: $i )).

thf(def_plusfrt,definition,
    ( plusfrt
    = ( d_Sigma @ frac
      @ ^ [X0: $i] :
          ( d_Sigma @ frac
          @ ^ [X1: $i] :
              ( ratof @ ( n_pf @ X0 @ X1 ) ) ) ) )).

thf(typ_rt_pl,type,(
    rt_pl: $i > $i > $i )).

thf(def_rt_pl,definition,
    ( rt_pl
    = ( ^ [X0: $i,X1: $i] :
          ( indrat @ X0 @ X1 @ rat @ plusfrt ) ) )).

thf(satz92,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_is @ ( rt_pl @ X0 @ X1 ) @ ( rt_pl @ X1 @ X0 ) ) ) )).

thf(satz93,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( rt_is @ ( rt_pl @ ( rt_pl @ X0 @ X1 ) @ X2 ) @ ( rt_pl @ X0 @ ( rt_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz94,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_more @ ( rt_pl @ X0 @ X1 ) @ X0 ) ) )).

thf(satz94a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_less @ X0 @ ( rt_pl @ X0 @ X1 ) ) ) )).

thf(satz95,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ X0 @ X1 )
               => ( rt_more @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz96b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ X0 @ X1 )
               => ( rt_is @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz96c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( rt_less @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz96d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ X0 @ X1 )
               => ( rt_more @ ( rt_pl @ X2 @ X0 ) @ ( rt_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz96e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ X0 @ X1 )
               => ( rt_is @ ( rt_pl @ X2 @ X0 ) @ ( rt_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz96f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( rt_less @ ( rt_pl @ X2 @ X0 ) @ ( rt_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz97a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X2 ) )
               => ( rt_more @ X0 @ X1 ) ) ) ) )).

thf(satz97b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X2 ) )
               => ( rt_is @ X0 @ X1 ) ) ) ) )).

thf(satz97c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X2 ) )
               => ( rt_less @ X0 @ X1 ) ) ) ) )).

thf(satz98,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_more @ X0 @ X1 )
                   => ( ( rt_more @ X2 @ X3 )
                     => ( rt_more @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz98a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_less @ X0 @ X1 )
                   => ( ( rt_less @ X2 @ X3 )
                     => ( rt_less @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz99a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_moreis @ X0 @ X1 )
                   => ( ( rt_more @ X2 @ X3 )
                     => ( rt_more @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz99b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_more @ X0 @ X1 )
                   => ( ( rt_moreis @ X2 @ X3 )
                     => ( rt_more @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz99c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_lessis @ X0 @ X1 )
                   => ( ( rt_less @ X2 @ X3 )
                     => ( rt_less @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz99d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_less @ X0 @ X1 )
                   => ( ( rt_lessis @ X2 @ X3 )
                     => ( rt_less @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz100,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_moreis @ X0 @ X1 )
                   => ( ( rt_moreis @ X2 @ X3 )
                     => ( rt_moreis @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz100a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_lessis @ X0 @ X1 )
                   => ( ( rt_lessis @ X2 @ X3 )
                     => ( rt_lessis @ ( rt_pl @ X0 @ X2 ) @ ( rt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz101a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_some
              @ ^ [X2: $i] :
                  ( rt_is @ ( rt_pl @ X1 @ X2 ) @ X0 ) ) ) ) )).

thf(satz101b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_is @ ( rt_pl @ X1 @ X2 ) @ X0 )
                   => ( ( rt_is @ ( rt_pl @ X1 @ X3 ) @ X0 )
                     => ( rt_is @ X2 @ X3 ) ) ) ) ) ) )).

thf(satz101,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_one
              @ ^ [X2: $i] :
                  ( rt_is @ ( rt_pl @ X1 @ X2 ) @ X0 ) ) ) ) )).

thf(typ_rt_mn,type,(
    rt_mn: $i > $i > $i )).

thf(def_rt_mn,definition,
    ( rt_mn
    = ( ^ [X0: $i,X1: $i] :
          ( ind @ rat
          @ ^ [X2: $i] :
              ( rt_is @ ( rt_pl @ X1 @ X2 ) @ X0 ) ) ) )).

thf(satz101c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_is @ ( rt_pl @ X1 @ ( rt_mn @ X0 @ X1 ) ) @ X0 ) ) ) )).

thf(satz101d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_is @ X0 @ ( rt_pl @ X1 @ ( rt_mn @ X0 @ X1 ) ) ) ) ) )).

thf(satz101e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_is @ ( rt_pl @ ( rt_mn @ X0 @ X1 ) @ X1 ) @ X0 ) ) ) )).

thf(satz101f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_is @ X0 @ ( rt_pl @ ( rt_mn @ X0 @ X1 ) @ X1 ) ) ) ) )).

thf(satz101g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ X0 @ X1 )
               => ( ( rt_is @ ( rt_pl @ X1 @ X2 ) @ X0 )
                 => ( rt_is @ X2 @ ( rt_mn @ X0 @ X1 ) ) ) ) ) ) )).

thf(typ_timesfrt,type,(
    timesfrt: $i )).

thf(def_timesfrt,definition,
    ( timesfrt
    = ( d_Sigma @ frac
      @ ^ [X0: $i] :
          ( d_Sigma @ frac
          @ ^ [X1: $i] :
              ( ratof @ ( n_tf @ X0 @ X1 ) ) ) ) )).

thf(typ_rt_ts,type,(
    rt_ts: $i > $i > $i )).

thf(def_rt_ts,definition,
    ( rt_ts
    = ( ^ [X0: $i,X1: $i] :
          ( indrat @ X0 @ X1 @ rat @ timesfrt ) ) )).

thf(satz102,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_is @ ( rt_ts @ X0 @ X1 ) @ ( rt_ts @ X1 @ X0 ) ) ) )).

thf(satz103,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( rt_is @ ( rt_ts @ ( rt_ts @ X0 @ X1 ) @ X2 ) @ ( rt_ts @ X0 @ ( rt_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz104,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( rt_is @ ( rt_ts @ X0 @ ( rt_pl @ X1 @ X2 ) ) @ ( rt_pl @ ( rt_ts @ X0 @ X1 ) @ ( rt_ts @ X0 @ X2 ) ) ) ) ) )).

thf(satz105a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ X0 @ X1 )
               => ( rt_more @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz105b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ X0 @ X1 )
               => ( rt_is @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz105c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( rt_less @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz105d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ X0 @ X1 )
               => ( rt_more @ ( rt_ts @ X2 @ X0 ) @ ( rt_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz105e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ X0 @ X1 )
               => ( rt_is @ ( rt_ts @ X2 @ X0 ) @ ( rt_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz105f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( rt_less @ ( rt_ts @ X2 @ X0 ) @ ( rt_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz106a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X2 ) )
               => ( rt_more @ X0 @ X1 ) ) ) ) )).

thf(satz106b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X2 ) )
               => ( rt_is @ X0 @ X1 ) ) ) ) )).

thf(satz106c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X2 ) )
               => ( rt_less @ X0 @ X1 ) ) ) ) )).

thf(satz107,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_more @ X0 @ X1 )
                   => ( ( rt_more @ X2 @ X3 )
                     => ( rt_more @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz107a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_less @ X0 @ X1 )
                   => ( ( rt_less @ X2 @ X3 )
                     => ( rt_less @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz108a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_moreis @ X0 @ X1 )
                   => ( ( rt_more @ X2 @ X3 )
                     => ( rt_more @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz108b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_more @ X0 @ X1 )
                   => ( ( rt_moreis @ X2 @ X3 )
                     => ( rt_more @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz108c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_lessis @ X0 @ X1 )
                   => ( ( rt_less @ X2 @ X3 )
                     => ( rt_less @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz108d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_less @ X0 @ X1 )
                   => ( ( rt_lessis @ X2 @ X3 )
                     => ( rt_less @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz109,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_moreis @ X0 @ X1 )
                   => ( ( rt_moreis @ X2 @ X3 )
                     => ( rt_moreis @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz109a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_lessis @ X0 @ X1 )
                   => ( ( rt_lessis @ X2 @ X3 )
                     => ( rt_lessis @ ( rt_ts @ X0 @ X2 ) @ ( rt_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz110a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_some
            @ ^ [X2: $i] :
                ( rt_is @ ( rt_ts @ X1 @ X2 ) @ X0 ) ) ) )).

thf(satz110b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ rat )
                @ ^ [X3: $i] :
                    ( ( rt_is @ ( rt_ts @ X1 @ X2 ) @ X0 )
                   => ( ( rt_is @ ( rt_ts @ X1 @ X3 ) @ X0 )
                     => ( rt_is @ X2 @ X3 ) ) ) ) ) ) )).

thf(satz110,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_one
            @ ^ [X2: $i] :
                ( rt_is @ ( rt_ts @ X1 @ X2 ) @ X0 ) ) ) )).

thf(satz111a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( moref @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) )
           => ( d_29_ii @ X0 @ X1 ) ) ) )).

thf(satz111b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( n_eq @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) )
           => ( n_is @ X0 @ X1 ) ) ) )).

thf(satz111c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( lessf @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) )
           => ( iii @ X0 @ X1 ) ) ) )).

thf(satz111d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ X0 @ X1 )
           => ( moref @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) ) ) ) )).

thf(satz111e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( n_is @ X0 @ X1 )
           => ( n_eq @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) ) ) ) )).

thf(satz111f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X0 @ X1 )
           => ( lessf @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) ) ) ) )).

thf(typ_natprop,type,(
    natprop: $i > $i > $o )).

thf(def_natprop,definition,
    ( natprop
    = ( ^ [X0: $i,X1: $i] :
          ( inf @ ( n_fr @ X1 @ n_1 ) @ ( class @ X0 ) ) ) )).

thf(typ_natrt,type,(
    natrt: $i > $o )).

thf(def_natrt,definition,
    ( natrt
    = ( ^ [X0: $i] :
          ( n_some @ ( natprop @ X0 ) ) ) )).

thf(satz111g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( ( natrt @ X0 )
       => ( n_one @ ( natprop @ X0 ) ) ) )).

thf(typ_nofrt,type,(
    nofrt: $i > $i )).

thf(def_nofrt,definition,
    ( nofrt
    = ( ^ [X0: $i] :
          ( ind @ nat @ ( natprop @ X0 ) ) ) )).

thf(typ_rtofn,type,(
    rtofn: $i > $i )).

thf(def_rtofn,definition,
    ( rtofn
    = ( ^ [X0: $i] :
          ( ratof @ ( n_fr @ X0 @ n_1 ) ) ) )).

thf(satz112a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_eq @ ( n_pf @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) ) @ ( n_fr @ ( n_pl @ X0 @ X1 ) @ n_1 ) ) ) )).

thf(satz112b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_eq @ ( n_tf @ ( n_fr @ X0 @ n_1 ) @ ( n_fr @ X1 @ n_1 ) ) @ ( n_fr @ ( n_ts @ X0 @ X1 ) @ n_1 ) ) ) )).

thf(satz112c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( ( natrt @ X0 )
       => ( all_of
          @ ^ [X1: $i] :
              ( in @ X1 @ rat )
          @ ^ [X1: $i] :
              ( ( natrt @ X1 )
             => ( inf @ ( n_fr @ ( n_pl @ ( nofrt @ X0 ) @ ( nofrt @ X1 ) ) @ n_1 ) @ ( class @ ( rt_pl @ X0 @ X1 ) ) ) ) ) ) )).

thf(satz112d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( ( natrt @ X0 )
       => ( all_of
          @ ^ [X1: $i] :
              ( in @ X1 @ rat )
          @ ^ [X1: $i] :
              ( ( natrt @ X1 )
             => ( natrt @ ( rt_pl @ X0 @ X1 ) ) ) ) ) )).

thf(satz112e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( ( natrt @ X0 )
       => ( all_of
          @ ^ [X1: $i] :
              ( in @ X1 @ rat )
          @ ^ [X1: $i] :
              ( ( natrt @ X1 )
             => ( inf @ ( n_fr @ ( n_ts @ ( nofrt @ X0 ) @ ( nofrt @ X1 ) ) @ n_1 ) @ ( class @ ( rt_ts @ X0 @ X1 ) ) ) ) ) ) )).

thf(satz112f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( ( natrt @ X0 )
       => ( all_of
          @ ^ [X1: $i] :
              ( in @ X1 @ rat )
          @ ^ [X1: $i] :
              ( ( natrt @ X1 )
             => ( natrt @ ( rt_ts @ X0 @ X1 ) ) ) ) ) )).

thf(satz112g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( ( natrt @ X0 )
       => ( all_of
          @ ^ [X1: $i] :
              ( in @ X1 @ rat )
          @ ^ [X1: $i] :
              ( ( natrt @ X1 )
             => ( ( rt_more @ X0 @ X1 )
               => ( natrt @ ( rt_mn @ X0 @ X1 ) ) ) ) ) ) )).

thf(satz112h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( rt_is @ ( rt_pl @ ( rtofn @ X0 ) @ ( rtofn @ X1 ) ) @ ( rtofn @ ( n_pl @ X0 @ X1 ) ) ) ) )).

thf(satz112j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( rt_is @ ( rt_ts @ ( rtofn @ X0 ) @ ( rtofn @ X1 ) ) @ ( rtofn @ ( n_ts @ X0 @ X1 ) ) ) ) )).

thf(typ_natt,type,(
    natt: $i )).

thf(def_natt,definition,
    ( natt
    = ( d_Sep @ rat @ natrt ) )).

thf(typ_ntofrt,type,(
    ntofrt: $i > $i )).

thf(def_ntofrt,definition,
    ( ntofrt
    = ( out @ rat @ natrt ) )).

thf(typ_nt_is,type,(
    nt_is: $i > $i > $o )).

thf(def_nt_is,definition,
    ( nt_is
    = ( e_is @ natt ) )).

thf(typ_nt_nis,type,(
    nt_nis: $i > $i > $o )).

thf(def_nt_nis,definition,
    ( nt_nis
    = ( ^ [X0: $i,X1: $i] :
          ( d_not @ ( nt_is @ X0 @ X1 ) ) ) )).

thf(typ_nt_all,type,(
    nt_all: ( $i > $o ) > $o )).

thf(def_nt_all,definition,
    ( nt_all
    = ( all @ natt ) )).

thf(typ_nt_some,type,(
    nt_some: ( $i > $o ) > $o )).

thf(def_nt_some,definition,
    ( nt_some
    = ( l_some @ natt ) )).

thf(typ_nt_one,type,(
    nt_one: ( $i > $o ) > $o )).

thf(def_nt_one,definition,
    ( nt_one
    = ( one @ natt ) )).

thf(typ_nt_in,type,(
    nt_in: $i > $i > $o )).

thf(def_nt_in,definition,
    ( nt_in
    = ( esti @ natt ) )).

thf(typ_rtofnt,type,(
    rtofnt: $i > $i )).

thf(def_rtofnt,definition,
    ( rtofnt
    = ( e_in @ rat @ natrt ) )).

thf(typ_ntofn,type,(
    ntofn: $i > $i )).

thf(def_ntofn,definition,
    ( ntofn
    = ( ^ [X0: $i] :
          ( ntofrt @ ( rtofn @ X0 ) ) ) )).

thf(typ_nofnt,type,(
    nofnt: $i > $i )).

thf(def_nofnt,definition,
    ( nofnt
    = ( ^ [X0: $i] :
          ( nofrt @ ( rtofnt @ X0 ) ) ) )).

thf(typ_nt_1t,type,(
    nt_1t: $i )).

thf(def_nt_1t,definition,
    ( nt_1t
    = ( ntofn @ n_1 ) )).

thf(typ_suct,type,(
    suct: $i )).

thf(def_suct,definition,
    ( suct
    = ( d_Sigma @ natt
      @ ^ [X0: $i] :
          ( ntofn @ ( ordsucc @ ( nofnt @ X0 ) ) ) ) )).

thf(satz113a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( nt_nis @ ( ap @ suct @ X0 ) @ nt_1t ) )).

thf(satz113b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ natt )
        @ ^ [X1: $i] :
            ( ( nt_is @ ( ap @ suct @ X0 ) @ ( ap @ suct @ X1 ) )
           => ( nt_is @ X0 @ X1 ) ) ) )).

thf(typ_nt_cond1,type,(
    nt_cond1: $i > $o )).

thf(def_nt_cond1,definition,
    ( nt_cond1
    = ( nt_in @ nt_1t ) )).

thf(typ_nt_cond2,type,(
    nt_cond2: $i > $o )).

thf(def_nt_cond2,definition,
    ( nt_cond2
    = ( ^ [X0: $i] :
          ( nt_all
          @ ^ [X1: $i] :
              ( imp @ ( nt_in @ X1 @ X0 ) @ ( nt_in @ ( ap @ suct @ X1 ) @ X0 ) ) ) ) )).

thf(typ_d_5113_prop1,type,(
    d_5113_prop1: $i > $i > $o )).

thf(def_d_5113_prop1,definition,
    ( d_5113_prop1
    = ( ^ [X0: $i,X1: $i] :
          ( nt_in @ ( ntofn @ X1 ) @ X0 ) ) )).

thf(satz113c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ ( power @ natt ) )
    @ ^ [X0: $i] :
        ( ( nt_cond1 @ X0 )
       => ( ( nt_cond2 @ X0 )
         => ( all_of
            @ ^ [X1: $i] :
                ( in @ X1 @ natt )
            @ ^ [X1: $i] :
                ( nt_in @ X1 @ X0 ) ) ) ) )).

thf(nt_satz1,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ natt )
        @ ^ [X1: $i] :
            ( ( nt_nis @ X0 @ X1 )
           => ( nt_nis @ ( ap @ suct @ X0 ) @ ( ap @ suct @ X1 ) ) ) ) )).

thf(typ_prop1t,type,(
    prop1t: $i > $o )).

thf(def_prop1t,definition,
    ( prop1t
    = ( ^ [X0: $i] :
          ( nt_all
          @ ^ [X1: $i] :
              ( nt_is @ ( ap @ X0 @ ( ap @ suct @ X1 ) ) @ ( ap @ suct @ ( ap @ X0 @ X1 ) ) ) ) ) )).

thf(typ_prop2t,type,(
    prop2t: $i > $i > $o )).

thf(def_prop2t,definition,
    ( prop2t
    = ( ^ [X0: $i,X1: $i] :
          ( d_and @ ( nt_is @ ( ap @ X1 @ nt_1t ) @ ( ap @ suct @ X0 ) ) @ ( prop1t @ X1 ) ) ) )).

thf(typ_d_54_prop2,type,(
    d_54_prop2: $i > $i > $o )).

thf(def_d_54_prop2,definition,
    ( d_54_prop2
    = ( ^ [X0: $i,X1: $i] :
          ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ ( ordsucc @ ( nofnt @ X0 ) ) ) @ ( d_24_prop1 @ X1 ) ) ) )).

thf(typ_d_54_g,type,(
    d_54_g: $i > $i )).

thf(def_d_54_g,definition,
    ( d_54_g
    = ( ^ [X0: $i] :
          ( d_Sigma @ nat
          @ ^ [X1: $i] :
              ( nofnt @ ( ap @ X0 @ ( ntofn @ X1 ) ) ) ) ) )).

thf(typ_d_54_gt,type,(
    d_54_gt: $i > $i )).

thf(def_d_54_gt,definition,
    ( d_54_gt
    = ( ^ [X0: $i] :
          ( d_Sigma @ natt
          @ ^ [X1: $i] :
              ( ntofn @ ( ap @ X0 @ ( nofnt @ X1 ) ) ) ) ) )).

thf(nt_satz4,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( one
        @ ( d_Pi @ natt
          @ ^ [X1: $i] : natt )
        @ ^ [X1: $i] :
            ( d_and @ ( nt_is @ ( ap @ X1 @ nt_1t ) @ ( ap @ suct @ X0 ) )
            @ ( nt_all
              @ ^ [X2: $i] :
                  ( nt_is @ ( ap @ X1 @ ( ap @ suct @ X2 ) ) @ ( ap @ suct @ ( ap @ X1 @ X2 ) ) ) ) ) ) )).

thf(typ_nt_pl,type,(
    nt_pl: $i > $i > $i )).

thf(def_nt_pl,definition,
    ( nt_pl
    = ( ^ [X0: $i,X1: $i] :
          ( ntofrt @ ( rt_pl @ ( rtofnt @ X0 ) @ ( rtofnt @ X1 ) ) ) ) )).

thf(nt_satz5,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ natt )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ natt )
            @ ^ [X2: $i] :
                ( nt_is @ ( nt_pl @ ( nt_pl @ X0 @ X1 ) @ X2 ) @ ( nt_pl @ X0 @ ( nt_pl @ X1 @ X2 ) ) ) ) ) )).

thf(typ_nt_diffprop,type,(
    nt_diffprop: $i > $i > $i > $o )).

thf(def_nt_diffprop,definition,
    ( nt_diffprop
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( nt_is @ X0 @ ( nt_pl @ X1 @ X2 ) ) ) )).

thf(typ_iit,type,(
    iit: $i > $i > $o )).

thf(def_iit,definition,
    ( iit
    = ( ^ [X0: $i,X1: $i] :
          ( nt_some @ ( nt_diffprop @ X0 @ X1 ) ) ) )).

thf(typ_iiit,type,(
    iiit: $i > $i > $o )).

thf(def_iiit,definition,
    ( iiit
    = ( ^ [X0: $i,X1: $i] :
          ( nt_some @ ( nt_diffprop @ X1 @ X0 ) ) ) )).

thf(typ_d_59_i,type,(
    d_59_i: $i > $i > $o )).

thf(def_d_59_i,definition,
    ( d_59_i
    = ( ^ [X0: $i,X1: $i] :
          ( n_is @ ( nofnt @ X0 ) @ ( nofnt @ X1 ) ) ) )).

thf(typ_d_59_ii,type,(
    d_59_ii: $i > $i > $o )).

thf(def_d_59_ii,definition,
    ( d_59_ii
    = ( ^ [X0: $i,X1: $i] :
          ( n_some @ ( diffprop @ ( nofnt @ X0 ) @ ( nofnt @ X1 ) ) ) ) )).

thf(typ_d_59_iii,type,(
    d_59_iii: $i > $i > $o )).

thf(def_d_59_iii,definition,
    ( d_59_iii
    = ( ^ [X0: $i,X1: $i] :
          ( n_some @ ( diffprop @ ( nofnt @ X1 ) @ ( nofnt @ X0 ) ) ) ) )).

thf(nt_satz9,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ natt )
        @ ^ [X1: $i] :
            ( orec3 @ ( nt_is @ X0 @ X1 )
            @ ( nt_some
              @ ^ [X2: $i] :
                  ( nt_is @ X0 @ ( nt_pl @ X1 @ X2 ) ) )
            @ ( nt_some
              @ ^ [X2: $i] :
                  ( nt_is @ X1 @ ( nt_pl @ X0 @ X2 ) ) ) ) ) )).

thf(typ_nt_more,type,(
    nt_more: $i > $i > $o )).

thf(def_nt_more,definition,
    ( nt_more
    = ( ^ [X0: $i,X1: $i] :
          ( rt_more @ ( rtofnt @ X0 ) @ ( rtofnt @ X1 ) ) ) )).

thf(typ_nt_less,type,(
    nt_less: $i > $i > $o )).

thf(def_nt_less,definition,
    ( nt_less
    = ( ^ [X0: $i,X1: $i] :
          ( rt_less @ ( rtofnt @ X0 ) @ ( rtofnt @ X1 ) ) ) )).

thf(typ_nt_moreis,type,(
    nt_moreis: $i > $i > $o )).

thf(def_nt_moreis,definition,
    ( nt_moreis
    = ( ^ [X0: $i,X1: $i] :
          ( rt_moreis @ ( rtofnt @ X0 ) @ ( rtofnt @ X1 ) ) ) )).

thf(typ_nt_lessis,type,(
    nt_lessis: $i > $i > $o )).

thf(def_nt_lessis,definition,
    ( nt_lessis
    = ( ^ [X0: $i,X1: $i] :
          ( rt_lessis @ ( rtofnt @ X0 ) @ ( rtofnt @ X1 ) ) ) )).

thf(nt_satz15,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ natt )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ natt )
            @ ^ [X2: $i] :
                ( ( nt_less @ X0 @ X1 )
               => ( ( nt_less @ X1 @ X2 )
                 => ( nt_less @ X0 @ X2 ) ) ) ) ) )).

thf(nt_satz21,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ natt )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ natt )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ natt )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ natt )
                @ ^ [X3: $i] :
                    ( ( nt_more @ X0 @ X1 )
                   => ( ( nt_more @ X2 @ X3 )
                     => ( nt_more @ ( nt_pl @ X0 @ X2 ) @ ( nt_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(typ_nt_lb,type,(
    nt_lb: ( $i > $o ) > $i > $o )).

thf(def_nt_lb,definition,
    ( nt_lb
    = ( ^ [X0: ( $i > $o ),X1: $i] :
          ( nt_all
          @ ^ [X2: $i] :
              ( imp @ ( X0 @ X2 ) @ ( nt_lessis @ X1 @ X2 ) ) ) ) )).

thf(typ_nt_min,type,(
    nt_min: ( $i > $o ) > $i > $o )).

thf(def_nt_min,definition,
    ( nt_min
    = ( ^ [X0: ( $i > $o ),X1: $i] :
          ( d_and @ ( nt_lb @ X0 @ X1 ) @ ( X0 @ X1 ) ) ) )).

thf(typ_d_527_q,type,(
    d_527_q: ( $i > $o ) > $i > $o )).

thf(def_d_527_q,definition,
    ( d_527_q
    = ( ^ [X0: ( $i > $o ),X1: $i] :
          ( X0 @ ( ntofn @ X1 ) ) ) )).

thf(nt_satz27,axiom,(
    ! [X0: ( $i > $o )] :
      ( ( nt_some @ X0 )
     => ( nt_some @ ( nt_min @ X0 ) ) ) )).

thf(typ_d_1rt,type,(
    d_1rt: $i )).

thf(def_d_1rt,definition,
    ( d_1rt
    = ( rtofn @ n_1 ) )).

thf(satz114,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( rt_is @ ( rt_ts @ ( rtofn @ ( den @ X0 ) ) @ ( ratof @ X0 ) ) @ ( rtofn @ ( num @ X0 ) ) ) )).

thf(satz114a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( rt_is @ ( rt_ts @ ( rtofn @ X1 ) @ ( ratof @ ( n_fr @ X0 @ X1 ) ) ) @ ( rtofn @ X0 ) ) ) )).

thf(typ_rt_ov,type,(
    rt_ov: $i > $i > $i )).

thf(def_rt_ov,definition,
    ( rt_ov
    = ( ^ [X0: $i,X1: $i] :
          ( ind @ rat
          @ ^ [X2: $i] :
              ( rt_is @ ( rt_ts @ X1 @ X2 ) @ X0 ) ) ) )).

thf(satz110c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_is @ ( rt_ts @ X1 @ ( rt_ov @ X0 @ X1 ) ) @ X0 ) ) )).

thf(satz110d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_is @ X0 @ ( rt_ts @ X1 @ ( rt_ov @ X0 @ X1 ) ) ) ) )).

thf(satz110e,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 @ X0 @ X1 ) @ X1 ) @ X0 ) ) )).

thf(satz110f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_is @ X0 @ ( rt_ts @ ( rt_ov @ X0 @ X1 ) @ X1 ) ) ) )).

thf(satz110g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ ( rt_ts @ X1 @ X2 ) @ X0 )
               => ( rt_is @ X2 @ ( rt_ov @ X0 @ X1 ) ) ) ) ) )).

thf(satz114b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( rt_is @ ( ratof @ ( n_fr @ X0 @ X1 ) ) @ ( rt_ov @ ( rtofn @ X0 ) @ ( rtofn @ X1 ) ) ) ) )).

thf(satz114c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( rt_is @ ( rt_ov @ ( rtofn @ X0 ) @ ( rtofn @ X1 ) ) @ ( ratof @ ( n_fr @ X0 @ X1 ) ) ) ) )).

thf(satz115,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( n_some
            @ ^ [X2: $i] :
                ( rt_more @ ( rt_ts @ ( rtofn @ X2 ) @ X0 ) @ X1 ) ) ) )).

thf(satz115a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( rt_some
            @ ^ [X2: $i] :
                ( d_and @ ( natrt @ X2 ) @ ( rt_more @ ( rt_ts @ X2 @ X0 ) @ X1 ) ) ) ) )).

thf(typ_cutprop1a,type,(
    cutprop1a: $i > $o )).

thf(def_cutprop1a,definition,
    ( cutprop1a
    = ( nonempty @ rat ) )).

thf(typ_cutprop1b,type,(
    cutprop1b: $i > $o )).

thf(def_cutprop1b,definition,
    ( cutprop1b
    = ( ^ [X0: $i] :
          ( d_not
          @ ( rt_all
            @ ^ [X1: $i] :
                ( rt_in @ X1 @ X0 ) ) ) ) )).

thf(typ_cutprop1,type,(
    cutprop1: $i > $o )).

thf(def_cutprop1,definition,
    ( cutprop1
    = ( ^ [X0: $i] :
          ( d_and @ ( cutprop1a @ X0 ) @ ( cutprop1b @ X0 ) ) ) )).

thf(typ_cutprop2a,type,(
    cutprop2a: $i > $i > $o )).

thf(def_cutprop2a,definition,
    ( cutprop2a
    = ( ^ [X0: $i,X1: $i] :
          ( rt_all
          @ ^ [X2: $i] :
              ( imp @ ( d_not @ ( rt_in @ X2 @ X0 ) ) @ ( rt_less @ X1 @ X2 ) ) ) ) )).

thf(typ_cutprop2,type,(
    cutprop2: $i > $o )).

thf(def_cutprop2,definition,
    ( cutprop2
    = ( ^ [X0: $i] :
          ( rt_all
          @ ^ [X1: $i] :
              ( imp @ ( rt_in @ X1 @ X0 ) @ ( cutprop2a @ X0 @ X1 ) ) ) ) )).

thf(typ_ubprop,type,(
    ubprop: $i > $i > $i > $o )).

thf(def_ubprop,definition,
    ( ubprop
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( imp @ ( rt_in @ X2 @ X0 ) @ ( rt_moreis @ X1 @ X2 ) ) ) )).

thf(typ_rt_ub,type,(
    rt_ub: $i > $i > $o )).

thf(def_rt_ub,definition,
    ( rt_ub
    = ( ^ [X0: $i,X1: $i] :
          ( rt_all @ ( ubprop @ X0 @ X1 ) ) ) )).

thf(typ_max,type,(
    max: $i > $i > $o )).

thf(def_max,definition,
    ( max
    = ( ^ [X0: $i,X1: $i] :
          ( d_and @ ( rt_ub @ X0 @ X1 ) @ ( rt_in @ X1 @ X0 ) ) ) )).

thf(typ_cutprop3,type,(
    cutprop3: $i > $o )).

thf(def_cutprop3,definition,
    ( cutprop3
    = ( ^ [X0: $i] :
          ( d_not @ ( rt_some @ ( max @ X0 ) ) ) ) )).

thf(typ_cutprop,type,(
    cutprop: $i > $o )).

thf(def_cutprop,definition,
    ( cutprop
    = ( ^ [X0: $i] :
          ( and3 @ ( cutprop1 @ X0 ) @ ( cutprop2 @ X0 ) @ ( cutprop3 @ X0 ) ) ) )).

thf(typ_iii1_lbprop,type,(
    iii1_lbprop: $i > $i > $i > $o )).

thf(def_iii1_lbprop,definition,
    ( iii1_lbprop
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( imp @ ( rt_in @ X2 @ X0 ) @ ( rt_lessis @ X1 @ X2 ) ) ) )).

thf(typ_rt_lb,type,(
    rt_lb: $i > $i > $o )).

thf(def_rt_lb,definition,
    ( rt_lb
    = ( ^ [X0: $i,X1: $i] :
          ( rt_all @ ( iii1_lbprop @ X0 @ X1 ) ) ) )).

thf(typ_rt_min,type,(
    rt_min: $i > $i > $o )).

thf(def_rt_min,definition,
    ( rt_min
    = ( ^ [X0: $i,X1: $i] :
          ( d_and @ ( rt_lb @ X0 @ X1 ) @ ( rt_in @ X1 @ X0 ) ) ) )).

thf(typ_cut,type,(
    cut: $i )).

thf(def_cut,definition,
    ( cut
    = ( d_Sep @ ( power @ rat ) @ cutprop ) )).

thf(typ_lcl,type,(
    lcl: $i > $i )).

thf(def_lcl,definition,
    ( lcl
    = ( e_in @ ( power @ rat ) @ cutprop ) )).

thf(typ_lrt,type,(
    lrt: $i > $i > $o )).

thf(def_lrt,definition,
    ( lrt
    = ( ^ [X0: $i,X1: $i] :
          ( rt_in @ X1 @ ( lcl @ X0 ) ) ) )).

thf(typ_urt,type,(
    urt: $i > $i > $o )).

thf(def_urt,definition,
    ( urt
    = ( ^ [X0: $i,X1: $i] :
          ( d_not @ ( rt_in @ X1 @ ( lcl @ X0 ) ) ) ) )).

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