TPTP Axioms File: NUM007^2.ax
%------------------------------------------------------------------------------
% File : NUM007^2 : TPTP v8.2.0. 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 ( 81 unt; 81 typ; 81 def)
% Number of atoms : 1282 ( 81 equ; 0 cnn)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 2535 ( 0 ~; 0 |; 0 &;2417 @)
% ( 0 <=>; 118 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg;2417 nst)
% Number of types : 2 ( 0 usr)
% Number of type conns : 141 ( 141 >; 0 *; 0 +; 0 <<)
% Number of symbols : 132 ( 131 usr; 58 con; 0-4 aty)
% Number of variables : 723 ( 722 ^ 1 !; 0 ?; 723 :)
% 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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