## 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      :

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