## TPTP Axioms File: NUM007^4.ax

```%------------------------------------------------------------------------------
% File     : NUM007^4 : TPTP v7.5.0. Released v7.1.0.
% Domain   : Number Theory
% Axioms   : Grundlagen chapter 4
% Version  : [Bro17] axioms : Especial.
% English  :

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

% Status   : Satisfiable
% Syntax   : Number of formulae    :  429 (   0 unit; 138 type; 138 defn)
%            Number of atoms       : 4053 ( 138 equality;1477 variable)
%            Maximal formula depth :   21 (   7 average)
%            Number of connectives : 3486 (   0   ~;   0   |;   0   &;3285   @)
%                                         (   0 <=>; 201  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  251 ( 251   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  200 ( 138   :;   0   =)
%            Number of variables   :  962 (  16 sgn;   0   !;   0   ?; 962   ^)
%                                         ( 962   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      :

%------------------------------------------------------------------------------
thf(typ_rp_eq,type,(
rp_eq: \$i > \$i > \$o )).

thf(def_rp_eq,definition,
( rp_eq
= ( ^ [X0: \$i,X1: \$i] :
( rp_is @ ( rp_pl @ ( stm @ X0 ) @ ( std @ X1 ) ) @ ( rp_pl @ ( stm @ X1 ) @ ( std @ X0 ) ) ) ) )).

thf(typ_posd,type,(
posd: \$i > \$o )).

thf(def_posd,definition,
( posd
= ( ^ [X0: \$i] :
( rp_more @ ( stm @ X0 ) @ ( std @ X0 ) ) ) )).

thf(typ_zero,type,(
zero: \$i > \$o )).

thf(def_zero,definition,
( zero
= ( ^ [X0: \$i] :
( rp_is @ ( stm @ X0 ) @ ( std @ X0 ) ) ) )).

thf(typ_negd,type,(
negd: \$i > \$o )).

thf(def_negd,definition,
( negd
= ( ^ [X0: \$i] :
( rp_less @ ( stm @ X0 ) @ ( std @ X0 ) ) ) )).

thf(typ_pdofrp,type,(
pdofrp: \$i > \$i )).

thf(def_pdofrp,definition,
( pdofrp
= ( ^ [X0: \$i] :
( rp_df @ ( rp_pl @ X0 @ d_1rp ) @ d_1rp ) ) )).

thf(typ_ndofrp,type,(
ndofrp: \$i > \$i )).

thf(def_ndofrp,definition,
( ndofrp
= ( ^ [X0: \$i] :
( rp_df @ d_1rp @ ( rp_pl @ X0 @ d_1rp ) ) ) )).

thf(typ_rpofpd,type,(
rpofpd: \$i > \$i )).

thf(def_rpofpd,definition,
( rpofpd
= ( ^ [X0: \$i] :
( rp_mn @ ( stm @ X0 ) @ ( std @ X0 ) ) ) )).

thf(typ_rpofnd,type,(
rpofnd: \$i > \$i )).

thf(def_rpofnd,definition,
( rpofnd
= ( ^ [X0: \$i] :
( rp_mn @ ( std @ X0 ) @ ( stm @ X0 ) ) ) )).

thf(typ_absd,type,(
absd: \$i > \$i )).

thf(def_absd,definition,
( absd
= ( ^ [X0: \$i] :
( ite @ ( negd @ X0 ) @ dif @ ( rp_df @ ( std @ X0 ) @ ( stm @ X0 ) ) @ X0 ) ) )).

thf(typ_mored,type,(
mored: \$i > \$i > \$o )).

thf(def_mored,definition,
( mored
= ( ^ [X0: \$i,X1: \$i] :
( rp_more @ ( rp_pl @ ( stm @ X0 ) @ ( std @ X1 ) ) @ ( rp_pl @ ( stm @ X1 ) @ ( std @ X0 ) ) ) ) )).

thf(typ_lessd,type,(
lessd: \$i > \$i > \$o )).

thf(def_lessd,definition,
( lessd
= ( ^ [X0: \$i,X1: \$i] :
( rp_less @ ( rp_pl @ ( stm @ X0 ) @ ( std @ X1 ) ) @ ( rp_pl @ ( stm @ X1 ) @ ( std @ X0 ) ) ) ) )).

thf(typ_rp_moreq,type,(
rp_moreq: \$i > \$i > \$o )).

thf(def_rp_moreq,definition,
( rp_moreq
= ( ^ [X0: \$i,X1: \$i] :
( l_or @ ( mored @ X0 @ X1 ) @ ( rp_eq @ X0 @ X1 ) ) ) )).

thf(typ_rp_lesseq,type,(
rp_lesseq: \$i > \$i > \$o )).

thf(def_rp_lesseq,definition,
( rp_lesseq
= ( ^ [X0: \$i,X1: \$i] :
( l_or @ ( lessd @ X0 @ X1 ) @ ( rp_eq @ X0 @ X1 ) ) ) )).

thf(typ_ratd,type,(
ratd: \$i > \$o )).

thf(def_ratd,definition,
( ratd
= ( ^ [X0: \$i] :
( ( d_not @ ( zero @ X0 ) )
=> ( ratrp @ ( rpofpd @ ( absd @ X0 ) ) ) ) ) )).

thf(typ_irratd,type,(
irratd: \$i > \$o )).

thf(def_irratd,definition,
( irratd
= ( ^ [X0: \$i] :
( d_not @ ( ratd @ X0 ) ) ) )).

thf(typ_natd,type,(
natd: \$i > \$o )).

thf(def_natd,definition,
( natd
= ( ^ [X0: \$i] :
( d_and @ ( posd @ X0 )
@ ( ( posd @ X0 )
=> ( natrp @ ( rpofpd @ X0 ) ) ) ) ) )).

thf(typ_pdofnt,type,(
pdofnt: \$i > \$i )).

thf(def_pdofnt,definition,
( pdofnt
= ( ^ [X0: \$i] :
( pdofrp @ ( rpofnt @ X0 ) ) ) )).

thf(typ_intd,type,(
intd: \$i > \$o )).

thf(def_intd,definition,
( intd
= ( ^ [X0: \$i] :
( l_or @ ( zero @ X0 ) @ ( natd @ ( absd @ X0 ) ) ) ) )).

thf(typ_rp_pd,type,(
rp_pd: \$i > \$i > \$i )).

thf(def_rp_pd,definition,
( rp_pd
= ( ^ [X0: \$i,X1: \$i] :
( rp_df @ ( rp_pl @ ( stm @ X0 ) @ ( stm @ X1 ) ) @ ( rp_pl @ ( std @ X0 ) @ ( std @ X1 ) ) ) ) )).

thf(typ_m0d,type,(
m0d: \$i > \$i )).

thf(def_m0d,definition,
( m0d
= ( ^ [X0: \$i] :
( rp_df @ ( std @ X0 ) @ ( stm @ X0 ) ) ) )).

thf(typ_rp_md,type,(
rp_md: \$i > \$i > \$i )).

thf(def_rp_md,definition,
( rp_md
= ( ^ [X0: \$i,X1: \$i] :
( rp_pd @ X0 @ ( m0d @ X1 ) ) ) )).

thf(typ_rp_td,type,(
rp_td: \$i > \$i > \$i )).

thf(def_rp_td,definition,
( rp_td
= ( ^ [X0: \$i,X1: \$i] :
( rp_df @ ( rp_pl @ ( rp_ts @ ( stm @ X0 ) @ ( stm @ X1 ) ) @ ( rp_ts @ ( std @ X0 ) @ ( std @ X1 ) ) ) @ ( rp_pl @ ( rp_ts @ ( stm @ X0 ) @ ( std @ X1 ) ) @ ( rp_ts @ ( std @ X0 ) @ ( stm @ X1 ) ) ) ) ) )).

thf(typ_d_1df,type,(
d_1df: \$i )).

thf(def_d_1df,definition,
( d_1df
= ( pdofrp @ d_1rp ) )).

thf(typ_p1p2,type,(
p1p2: \$i > \$i > \$o )).

thf(def_p1p2,definition,
( p1p2
= ( ^ [X0: \$i,X1: \$i] :
( d_and @ ( posd @ X0 ) @ ( posd @ X1 ) ) ) )).

thf(typ_p1n2,type,(
p1n2: \$i > \$i > \$o )).

thf(def_p1n2,definition,
( p1n2
= ( ^ [X0: \$i,X1: \$i] :
( d_and @ ( posd @ X0 ) @ ( negd @ X1 ) ) ) )).

thf(typ_n1p2,type,(
n1p2: \$i > \$i > \$o )).

thf(def_n1p2,definition,
( n1p2
= ( ^ [X0: \$i,X1: \$i] :
( d_and @ ( negd @ X0 ) @ ( posd @ X1 ) ) ) )).

thf(typ_n1n2,type,(
n1n2: \$i > \$i > \$o )).

thf(def_n1n2,definition,
( n1n2
= ( ^ [X0: \$i,X1: \$i] :
( d_and @ ( negd @ X0 ) @ ( negd @ X1 ) ) ) )).

thf(typ_arpi,type,(
arpi: \$i > \$i )).

thf(def_arpi,definition,
( arpi
= ( ^ [X0: \$i] :
( rp_ov @ d_1rp @ ( rpofpd @ X0 ) ) ) )).

thf(typ_iv4d_ai,type,(
iv4d_ai: \$i > \$i )).

thf(def_iv4d_ai,definition,
( iv4d_ai
= ( ^ [X0: \$i] :
( pdofrp @ ( arpi @ X0 ) ) ) )).

thf(typ_iv5d_2,type,(
iv5d_2: \$i )).

thf(def_iv5d_2,definition,
( iv5d_2
= ( rp_pl @ d_1rp @ d_1rp ) )).

thf(typ_rp1,type,(
rp1: \$i > \$i )).

thf(def_rp1,definition,
( rp1
= ( ^ [X0: \$i] :
( rp_pl @ X0 @ d_1rp ) ) )).

thf(typ_rp_in,type,(
rp_in: \$i > \$i > \$o )).

thf(def_rp_in,definition,
( rp_in
= ( esti @ cut ) )).

thf(typ_d_5p205_prop1,type,(
d_5p205_prop1: \$i > \$i > \$o )).

thf(def_d_5p205_prop1,definition,
( d_5p205_prop1
= ( ^ [X0: \$i,X1: \$i] :
( rp_all
@ ^ [X2: \$i] :
( ( rp_less @ X2 @ X1 )
=> ( rp_in @ X2 @ X0 ) ) ) ) )).

thf(typ_d_5p205_prop2,type,(
d_5p205_prop2: \$i > \$i > \$o )).

thf(def_d_5p205_prop2,definition,
( d_5p205_prop2
= ( ^ [X0: \$i,X1: \$i] :
( rp_all
@ ^ [X2: \$i] :
( ( rp_more @ X2 @ X1 )
=> ( rp_in @ X2 @ X0 ) ) ) ) )).

thf(typ_d_5p205_prop3,type,(
d_5p205_prop3: \$i > \$i > \$i > \$o )).

thf(def_d_5p205_prop3,definition,
( d_5p205_prop3
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( d_and @ ( d_5p205_prop1 @ X0 @ X2 ) @ ( d_5p205_prop2 @ X1 @ X2 ) ) ) )).

thf(typ_schnittprop,type,(
schnittprop: \$i > \$i > \$o )).

thf(def_schnittprop,definition,
( schnittprop
= ( ^ [X0: \$i,X1: \$i] :
( rp_some
@ ^ [X2: \$i] :
( d_and @ ( rp_in @ X2 @ X0 ) @ ( lrt @ X2 @ X1 ) ) ) ) )).

thf(typ_schnittset,type,(
schnittset: \$i > \$i )).

thf(def_schnittset,definition,
( schnittset
= ( ^ [X0: \$i] :
( d_Sep @ rat @ ( schnittprop @ X0 ) ) ) )).

thf(typ_snt,type,(
snt: \$i > \$i > \$i )).

thf(def_snt,definition,
( snt
= ( ^ [X0: \$i,X1: \$i] :
( cutof @ ( schnittset @ X0 ) ) ) )).

thf(typ_schnitt,type,(
schnitt: \$i > \$i > \$i )).

thf(def_schnitt,definition,
( schnitt
= ( ^ [X0: \$i,X1: \$i] :
( ind @ cut @ ( d_5p205_prop3 @ X0 @ X1 ) ) ) )).

thf(typ_srp,type,(
srp: \$i > \$i )).

thf(def_srp,definition,
( srp
= ( ^ [X0: \$i] :
( sqrt @ ( rpofpd @ X0 ) ) ) )).

thf(typ_d161_s,type,(
d161_s: \$i > \$i )).

thf(def_d161_s,definition,
( d161_s
= ( ^ [X0: \$i] :
( pdofrp @ ( srp @ X0 ) ) ) )).

thf(typ_apb1,type,(
apb1: \$i > \$i > \$i )).

thf(def_apb1,definition,
( apb1
= ( ^ [X0: \$i,X1: \$i] :
( rpofpd @ ( rp_pd @ X0 @ X1 ) ) ) )).

thf(typ_intd_b2,type,(
intd_b2: \$i > \$i )).

thf(def_intd_b2,definition,
( intd_b2
= ( ^ [X0: \$i] :
( rpofpd @ ( m0d @ X0 ) ) ) )).

thf(typ_intd_a3,type,(
intd_a3: \$i > \$i > \$i )).

thf(def_intd_a3,definition,
( intd_a3
= ( ^ [X0: \$i,X1: \$i] :
( rpofpd @ ( absd @ X0 ) ) ) )).

thf(typ_intd_b3,type,(
intd_b3: \$i > \$i > \$i )).

thf(def_intd_b3,definition,
( intd_b3
= ( ^ [X0: \$i,X1: \$i] :
( rpofpd @ ( absd @ X1 ) ) ) )).

thf(typ_atb3,type,(
atb3: \$i > \$i > \$i )).

thf(def_atb3,definition,
( atb3
= ( ^ [X0: \$i,X1: \$i] :
( rpofpd @ ( absd @ ( rp_td @ X0 @ X1 ) ) ) ) )).

thf(typ_r_inn,type,(
r_inn: \$i > \$i > \$o )).

thf(def_r_inn,definition,
( r_inn
= ( esti @ dif ) )).

thf(typ_real,type,(
real: \$i )).

thf(def_real,definition,
( real
= ( ect @ dif @ rp_eq ) )).

thf(typ_r_is,type,(
r_is: \$i > \$i > \$o )).

thf(def_r_is,definition,
( r_is
= ( e_is @ real ) )).

thf(typ_r_nis,type,(
r_nis: \$i > \$i > \$o )).

thf(def_r_nis,definition,
( r_nis
= ( ^ [X0: \$i,X1: \$i] :
( d_not @ ( r_is @ X0 @ X1 ) ) ) )).

thf(typ_r_some,type,(
r_some: ( \$i > \$o ) > \$o )).

thf(def_r_some,definition,
( r_some
= ( l_some @ real ) )).

thf(typ_r_all,type,(
r_all: ( \$i > \$o ) > \$o )).

thf(def_r_all,definition,
( r_all
= ( all @ real ) )).

thf(typ_r_one,type,(
r_one: ( \$i > \$o ) > \$o )).

thf(def_r_one,definition,
( r_one
= ( one @ real ) )).

thf(typ_r_in,type,(
r_in: \$i > \$i > \$o )).

thf(def_r_in,definition,
( r_in
= ( esti @ real ) )).

thf(typ_realof,type,(
realof: \$i > \$i )).

thf(def_realof,definition,
( realof
= ( ectelt @ dif @ rp_eq ) )).

thf(typ_r_class,type,(
r_class: \$i > \$i )).

thf(def_r_class,definition,
( r_class
= ( ecect @ dif @ rp_eq ) )).

thf(typ_r_fixf,type,(
r_fixf: \$i > \$i > \$o )).

thf(def_r_fixf,definition,
( r_fixf
= ( fixfu @ dif @ rp_eq ) )).

thf(typ_indreal,type,(
indreal: \$i > \$i > \$i > \$i )).

thf(def_indreal,definition,
( indreal
= ( indeq @ dif @ rp_eq ) )).

thf(typ_fixf2,type,(
fixf2: \$i > \$i > \$o )).

thf(def_fixf2,definition,
( fixf2
= ( fixfu2 @ dif @ rp_eq ) )).

thf(typ_indreal2,type,(
indreal2: \$i > \$i > \$i > \$i > \$i )).

thf(def_indreal2,definition,
( indreal2
= ( indeq2 @ dif @ rp_eq ) )).

thf(typ_r_0,type,(
r_0: \$i )).

thf(def_r_0,definition,
( r_0
= ( realof @ ( rp_df @ d_1rp @ d_1rp ) ) )).

thf(typ_propp,type,(
propp: \$i > \$i > \$o )).

thf(def_propp,definition,
( propp
= ( ^ [X0: \$i,X1: \$i] :
( d_and @ ( r_inn @ X1 @ ( r_class @ X0 ) ) @ ( posd @ X1 ) ) ) )).

thf(typ_pos,type,(
pos: \$i > \$o )).

thf(def_pos,definition,
( pos
= ( ^ [X0: \$i] :
( l_some @ dif @ ( propp @ X0 ) ) ) )).

thf(typ_propn,type,(
propn: \$i > \$i > \$o )).

thf(def_propn,definition,
( propn
= ( ^ [X0: \$i,X1: \$i] :
( d_and @ ( r_inn @ X1 @ ( r_class @ X0 ) ) @ ( negd @ X1 ) ) ) )).

thf(typ_neg,type,(
neg: \$i > \$o )).

thf(def_neg,definition,
( neg
= ( ^ [X0: \$i] :
( l_some @ dif @ ( propn @ X0 ) ) ) )).

thf(typ_pofrp,type,(
pofrp: \$i > \$i )).

thf(def_pofrp,definition,
( pofrp
= ( ^ [X0: \$i] :
( realof @ ( pdofrp @ X0 ) ) ) )).

thf(typ_nofrp,type,(
nofrp: \$i > \$i )).

thf(def_nofrp,definition,
( nofrp
= ( ^ [X0: \$i] :
( realof @ ( ndofrp @ X0 ) ) ) )).

thf(typ_ivr1_pr,type,(
ivr1_pr: \$i > \$i > \$i )).

thf(def_ivr1_pr,definition,
( ivr1_pr
= ( ^ [X0: \$i] : rpofpd ) )).

thf(typ_rpofp,type,(
rpofp: \$i > \$i )).

thf(def_rpofp,definition,
( rpofp
= ( ^ [X0: \$i] :
( ind @ cut
@ ^ [X1: \$i] :
( r_is @ X0 @ ( pofrp @ X1 ) ) ) ) )).

thf(typ_ivr1_nr,type,(
ivr1_nr: \$i > \$i > \$i )).

thf(def_ivr1_nr,definition,
( ivr1_nr
= ( ^ [X0: \$i] : rpofnd ) )).

thf(typ_rpofn,type,(
rpofn: \$i > \$i )).

thf(def_rpofn,definition,
( rpofn
= ( ^ [X0: \$i] :
( ind @ cut
@ ^ [X1: \$i] :
( r_is @ X0 @ ( nofrp @ X1 ) ) ) ) )).

thf(satz163,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ X0 @ X0 ) )).

thf(satz164,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ X0 @ X1 )
=> ( r_is @ X1 @ X0 ) ) ) )).

thf(satz165,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ X0 @ X1 )
=> ( ( r_is @ X1 @ X2 )
=> ( r_is @ X0 @ X2 ) ) ) ) ) )).

thf(typ_absdr,type,(
absdr: \$i )).

thf(def_absdr,definition,
( absdr
= ( d_Sigma @ dif
@ ^ [X0: \$i] :
( realof @ ( absd @ X0 ) ) ) )).

thf(typ_abs,type,(
abs: \$i > \$i )).

thf(def_abs,definition,
( abs
= ( indreal @ real @ absdr ) )).

thf(satz166a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( pos @ X0 )
=> ( pos @ ( abs @ X0 ) ) ) )).

thf(satz166b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( neg @ X0 )
=> ( pos @ ( abs @ X0 ) ) ) )).

thf(satz166c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( pos @ X0 )
=> ( ( pos @ X1 )
=> ( ( r_is @ ( abs @ X0 ) @ ( abs @ X1 ) )
=> ( r_is @ X0 @ X1 ) ) ) ) ) )).

thf(satz166d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( neg @ X0 )
=> ( ( neg @ X1 )
=> ( ( r_is @ ( abs @ X0 ) @ ( abs @ X1 ) )
=> ( r_is @ X0 @ X1 ) ) ) ) ) )).

thf(satz166e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( r_nis @ X0 @ r_0 )
=> ( pos @ ( abs @ X0 ) ) ) )).

thf(satz166f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( r_is @ X0 @ r_0 )
=> ( r_is @ ( abs @ X0 ) @ r_0 ) ) )).

thf(typ_r_more,type,(
r_more: \$i > \$i > \$o )).

thf(def_r_more,definition,
( r_more
= ( ^ [X0: \$i,X1: \$i] :
( l_some @ dif
@ ^ [X2: \$i] :
( l_some @ dif
@ ^ [X3: \$i] :
( and3 @ ( r_inn @ X2 @ ( r_class @ X0 ) ) @ ( r_inn @ X3 @ ( r_class @ X1 ) ) @ ( mored @ X2 @ X3 ) ) ) ) ) )).

thf(typ_ivr2_propm,type,(
ivr2_propm: \$i > \$i > \$i > \$i > \$o )).

thf(def_ivr2_propm,definition,
( ivr2_propm
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] :
( and3 @ ( r_inn @ X2 @ ( r_class @ X0 ) ) @ ( r_inn @ X3 @ ( r_class @ X1 ) ) @ ( mored @ X2 @ X3 ) ) ) )).

thf(typ_r_less,type,(
r_less: \$i > \$i > \$o )).

thf(def_r_less,definition,
( r_less
= ( ^ [X0: \$i,X1: \$i] :
( l_some @ dif
@ ^ [X2: \$i] :
( l_some @ dif
@ ^ [X3: \$i] :
( and3 @ ( r_inn @ X2 @ ( r_class @ X0 ) ) @ ( r_inn @ X3 @ ( r_class @ X1 ) ) @ ( lessd @ X2 @ X3 ) ) ) ) ) )).

thf(typ_ivr2_propl,type,(
ivr2_propl: \$i > \$i > \$i > \$i > \$o )).

thf(def_ivr2_propl,definition,
( ivr2_propl
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] :
( and3 @ ( r_inn @ X2 @ ( r_class @ X0 ) ) @ ( r_inn @ X3 @ ( r_class @ X1 ) ) @ ( lessd @ X2 @ X3 ) ) ) )).

thf(satz167,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( orec3 @ ( r_is @ X0 @ X1 ) @ ( r_more @ X0 @ X1 ) @ ( r_less @ X0 @ X1 ) ) ) )).

thf(satz167a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( or3 @ ( r_is @ X0 @ X1 ) @ ( r_more @ X0 @ X1 ) @ ( r_less @ X0 @ X1 ) ) ) )).

thf(satz167b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ec3 @ ( r_is @ X0 @ X1 ) @ ( r_more @ X0 @ X1 ) @ ( r_less @ X0 @ X1 ) ) ) )).

thf(typ_r_moreis,type,(
r_moreis: \$i > \$i > \$o )).

thf(def_r_moreis,definition,
( r_moreis
= ( ^ [X0: \$i,X1: \$i] :
( l_or @ ( r_more @ X0 @ X1 ) @ ( r_is @ X0 @ X1 ) ) ) )).

thf(typ_r_lessis,type,(
r_lessis: \$i > \$i > \$o )).

thf(def_r_lessis,definition,
( r_lessis
= ( ^ [X0: \$i,X1: \$i] :
( l_or @ ( r_less @ X0 @ X1 ) @ ( r_is @ X0 @ X1 ) ) ) )).

thf(satz168a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_moreis @ X0 @ X1 )
=> ( r_lessis @ X1 @ X0 ) ) ) )).

thf(satz168b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_lessis @ X0 @ X1 )
=> ( r_moreis @ X1 @ X0 ) ) ) )).

thf(satz167c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_moreis @ X0 @ X1 )
=> ( d_not @ ( r_less @ X0 @ X1 ) ) ) ) )).

thf(satz167d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_lessis @ X0 @ X1 )
=> ( d_not @ ( r_more @ X0 @ X1 ) ) ) ) )).

thf(satz167e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( d_not @ ( r_more @ X0 @ X1 ) )
=> ( r_lessis @ X0 @ X1 ) ) ) )).

thf(satz167f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( d_not @ ( r_less @ X0 @ X1 ) )
=> ( r_moreis @ X0 @ X1 ) ) ) )).

thf(satz167g,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_more @ X0 @ X1 )
=> ( d_not @ ( r_lessis @ X0 @ X1 ) ) ) ) )).

thf(satz167h,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_less @ X0 @ X1 )
=> ( d_not @ ( r_moreis @ X0 @ X1 ) ) ) ) )).

thf(satz167j,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( d_not @ ( r_moreis @ X0 @ X1 ) )
=> ( r_less @ X0 @ X1 ) ) ) )).

thf(satz167k,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( d_not @ ( r_lessis @ X0 @ X1 ) )
=> ( r_more @ X0 @ X1 ) ) ) )).

thf(satz169a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( pos @ X0 )
=> ( r_more @ X0 @ r_0 ) ) )).

thf(satz169b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( r_more @ X0 @ r_0 )
=> ( pos @ X0 ) ) )).

thf(satz169c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( neg @ X0 )
=> ( r_less @ X0 @ r_0 ) ) )).

thf(satz169d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( r_less @ X0 @ r_0 )
=> ( neg @ X0 ) ) )).

thf(satz170,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_moreis @ ( abs @ X0 ) @ r_0 ) )).

thf(satz170a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( d_not @ ( neg @ ( abs @ X0 ) ) ) )).

thf(satz171,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( r_less @ X1 @ X2 )
=> ( r_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz172a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_lessis @ X0 @ X1 )
=> ( ( r_less @ X1 @ X2 )
=> ( r_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz172b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( r_lessis @ X1 @ X2 )
=> ( r_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz172c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_moreis @ X0 @ X1 )
=> ( ( r_more @ X1 @ X2 )
=> ( r_more @ X0 @ X2 ) ) ) ) ) )).

thf(satz172d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X0 @ X1 )
=> ( ( r_moreis @ X1 @ X2 )
=> ( r_more @ X0 @ X2 ) ) ) ) ) )).

thf(satz173,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_lessis @ X0 @ X1 )
=> ( ( r_lessis @ X1 @ X2 )
=> ( r_lessis @ X0 @ X2 ) ) ) ) ) )).

thf(typ_ratrl,type,(
ratrl: \$i > \$o )).

thf(def_ratrl,definition,
( ratrl
= ( ^ [X0: \$i] :
( l_some @ dif
@ ^ [X1: \$i] :
( d_and @ ( r_inn @ X1 @ ( r_class @ X0 ) ) @ ( ratd @ X1 ) ) ) ) )).

thf(typ_irratrl,type,(
irratrl: \$i > \$o )).

thf(def_irratrl,definition,
( irratrl
= ( ^ [X0: \$i] :
( d_not @ ( ratrl @ X0 ) ) ) )).

thf(typ_natrl,type,(
natrl: \$i > \$o )).

thf(def_natrl,definition,
( natrl
= ( ^ [X0: \$i] :
( l_some @ dif
@ ^ [X1: \$i] :
( d_and @ ( r_inn @ X1 @ ( r_class @ X0 ) ) @ ( natd @ X1 ) ) ) ) )).

thf(typ_rlofnt,type,(
rlofnt: \$i > \$i )).

thf(def_rlofnt,definition,
( rlofnt
= ( ^ [X0: \$i] :
( realof @ ( pdofnt @ X0 ) ) ) )).

thf(typ_ivr2_x0,type,(
ivr2_x0: \$i > \$i > \$i )).

thf(def_ivr2_x0,definition,
( ivr2_x0
= ( ^ [X0: \$i,X1: \$i] :
( ntofrp @ ( ivr1_pr @ X0 @ X1 ) ) ) )).

thf(typ_ntofrl,type,(
ntofrl: \$i > \$i )).

thf(def_ntofrl,definition,
( ntofrl
= ( soft @ nat @ real @ ( d_Sigma @ nat @ rlofnt ) ) )).

thf(typ_ivr2_xn,type,(
ivr2_xn: \$i > \$i )).

thf(def_ivr2_xn,definition,
( ivr2_xn
= ( ^ [X0: \$i] :
( soft @ nat @ real @ ( d_Sigma @ nat @ rlofnt ) @ ( rlofnt @ X0 ) ) ) )).

thf(typ_intrl,type,(
intrl: \$i > \$o )).

thf(def_intrl,definition,
( intrl
= ( ^ [X0: \$i] :
( l_some @ dif
@ ^ [X1: \$i] :
( d_and @ ( r_inn @ X1 @ ( r_class @ X0 ) ) @ ( intd @ X1 ) ) ) ) )).

thf(satz174,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( intrl @ X0 )
=> ( ratrl @ X0 ) ) )).

thf(typ_plusdr,type,(
plusdr: \$i )).

thf(def_plusdr,definition,
( plusdr
= ( d_Sigma @ dif
@ ^ [X0: \$i] :
( d_Sigma @ dif
@ ^ [X1: \$i] :
( realof @ ( rp_pd @ X0 @ X1 ) ) ) ) )).

thf(typ_r_pl,type,(
r_pl: \$i > \$i > \$i )).

thf(def_r_pl,definition,
( r_pl
= ( indreal2 @ real @ plusdr ) )).

thf(satz175,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_pl @ X0 @ X1 ) @ ( r_pl @ X1 @ X0 ) ) ) )).

thf(typ_m0dr,type,(
m0dr: \$i )).

thf(def_m0dr,definition,
( m0dr
= ( d_Sigma @ dif
@ ^ [X0: \$i] :
( realof @ ( m0d @ X0 ) ) ) )).

thf(typ_r_m0,type,(
r_m0: \$i > \$i )).

thf(def_r_m0,definition,
( r_m0
= ( indreal @ real @ m0dr ) )).

thf(satz176a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( pos @ X0 )
=> ( neg @ ( r_m0 @ X0 ) ) ) )).

thf(satz176b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( r_is @ X0 @ r_0 )
=> ( r_is @ ( r_m0 @ X0 ) @ r_0 ) ) )).

thf(satz176c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( neg @ X0 )
=> ( pos @ ( r_m0 @ X0 ) ) ) )).

thf(satz176d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( neg @ ( r_m0 @ X0 ) )
=> ( pos @ X0 ) ) )).

thf(satz176e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( r_is @ ( r_m0 @ X0 ) @ r_0 )
=> ( r_is @ X0 @ r_0 ) ) )).

thf(satz176f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( ( pos @ ( r_m0 @ X0 ) )
=> ( neg @ X0 ) ) )).

thf(satz177,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ ( r_m0 @ ( r_m0 @ X0 ) ) @ X0 ) )).

thf(satz177a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ X0 @ ( r_m0 @ ( r_m0 @ X0 ) ) ) )).

thf(satz177b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ X0 @ ( r_m0 @ X1 ) )
=> ( r_is @ ( r_m0 @ X0 ) @ X1 ) ) ) )).

thf(satz177c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ X0 @ ( r_m0 @ X1 ) )
=> ( r_is @ X1 @ ( r_m0 @ X0 ) ) ) ) )).

thf(satz177d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ ( r_m0 @ X0 ) @ X1 )
=> ( r_is @ X0 @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz177e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ ( r_m0 @ X0 ) @ X1 )
=> ( r_is @ ( r_m0 @ X1 ) @ X0 ) ) ) )).

thf(satz178,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ ( abs @ ( r_m0 @ X0 ) ) @ ( abs @ X0 ) ) )).

thf(satz178a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ ( abs @ X0 ) @ ( abs @ ( r_m0 @ X0 ) ) ) )).

thf(satz179,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ ( r_pl @ X0 @ ( r_m0 @ X0 ) ) @ r_0 ) )).

thf(satz179a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ ( r_pl @ ( r_m0 @ X0 ) @ X0 ) @ r_0 ) )).

thf(satz180,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_m0 @ ( r_pl @ X0 @ X1 ) ) @ ( r_pl @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz180a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_pl @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) ) @ ( r_m0 @ ( r_pl @ X0 @ X1 ) ) ) ) )).

thf(typ_r_mn,type,(
r_mn: \$i > \$i > \$i )).

thf(def_r_mn,definition,
( r_mn
= ( ^ [X0: \$i,X1: \$i] :
( r_pl @ X0 @ ( r_m0 @ X1 ) ) ) )).

thf(satz181,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_m0 @ ( r_mn @ X0 @ X1 ) ) @ ( r_mn @ X1 @ X0 ) ) ) )).

thf(satz181a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_mn @ X0 @ X1 ) @ ( r_m0 @ ( r_mn @ X1 @ X0 ) ) ) ) )).

thf(satz182a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( pos @ ( r_mn @ X0 @ X1 ) )
=> ( r_more @ X0 @ X1 ) ) ) )).

thf(satz182b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ ( r_mn @ X0 @ X1 ) @ r_0 )
=> ( r_is @ X0 @ X1 ) ) ) )).

thf(satz182c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( neg @ ( r_mn @ X0 @ X1 ) )
=> ( r_less @ X0 @ X1 ) ) ) )).

thf(satz182d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_more @ X0 @ X1 )
=> ( pos @ ( r_mn @ X0 @ X1 ) ) ) ) )).

thf(satz182e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ X0 @ X1 )
=> ( r_is @ ( r_mn @ X0 @ X1 ) @ r_0 ) ) ) )).

thf(satz182f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_less @ X0 @ X1 )
=> ( neg @ ( r_mn @ X0 @ X1 ) ) ) ) )).

thf(satz183a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_more @ X0 @ X1 )
=> ( r_less @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz183b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ X0 @ X1 )
=> ( r_is @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz183c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_less @ X0 @ X1 )
=> ( r_more @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz183d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_less @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) )
=> ( r_more @ X0 @ X1 ) ) ) )).

thf(satz183e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) )
=> ( r_is @ X0 @ X1 ) ) ) )).

thf(satz183f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_more @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) )
=> ( r_less @ X0 @ X1 ) ) ) )).

thf(typ_d_3r184_prop1,type,(
d_3r184_prop1: \$i > \$i > \$i > \$o )).

thf(def_d_3r184_prop1,definition,
( d_3r184_prop1
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( and3 @ ( pos @ X1 ) @ ( pos @ X2 ) @ ( r_is @ X0 @ ( r_mn @ X1 @ X2 ) ) ) ) )).

thf(typ_d_3r184_prop2,type,(
d_3r184_prop2: \$i > \$i > \$o )).

thf(def_d_3r184_prop2,definition,
( d_3r184_prop2
= ( ^ [X0: \$i,X1: \$i] :
( r_some @ ( d_3r184_prop1 @ X0 @ X1 ) ) ) )).

thf(typ_d_3r184_prop3,type,(
d_3r184_prop3: \$i > \$o )).

thf(def_d_3r184_prop3,definition,
( d_3r184_prop3
= ( ^ [X0: \$i] :
( r_some @ ( d_3r184_prop2 @ X0 ) ) ) )).

thf(typ_prop1d,type,(
prop1d: \$i > \$i > \$i > \$o )).

thf(def_prop1d,definition,
( prop1d
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( and3 @ ( posd @ X1 ) @ ( posd @ X2 ) @ ( rp_eq @ X0 @ ( rp_md @ X1 @ X2 ) ) ) ) )).

thf(typ_prop2d,type,(
prop2d: \$i > \$i > \$o )).

thf(def_prop2d,definition,
( prop2d
= ( ^ [X0: \$i,X1: \$i] :
( l_some @ dif @ ( prop1d @ X0 @ X1 ) ) ) )).

thf(satz184,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_some
@ ^ [X1: \$i] :
( r_some
@ ^ [X2: \$i] :
( and3 @ ( pos @ X1 ) @ ( pos @ X2 ) @ ( r_is @ X0 @ ( r_mn @ X1 @ X2 ) ) ) ) ) )).

thf(satz185,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( r_is @ ( r_pl @ ( r_mn @ X0 @ X1 ) @ ( r_mn @ X2 @ X3 ) ) @ ( r_mn @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) )).

thf(satz186,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( r_is @ ( r_pl @ ( r_pl @ X0 @ X1 ) @ X2 ) @ ( r_pl @ X0 @ ( r_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz187a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_pl @ X1 @ ( r_mn @ X0 @ X1 ) ) @ X0 ) ) )).

thf(satz187b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_some
@ ^ [X2: \$i] :
( r_is @ ( r_pl @ X1 @ X2 ) @ X0 ) ) ) )).

thf(satz187c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ ( r_pl @ X1 @ X2 ) @ X0 )
=> ( r_is @ ( r_mn @ X0 @ X1 ) @ X2 ) ) ) ) )).

thf(satz187d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ ( r_pl @ X1 @ X2 ) @ X0 )
=> ( r_is @ X2 @ ( r_mn @ X0 @ X1 ) ) ) ) ) )).

thf(satz187e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ ( r_pl @ X2 @ X1 ) @ X0 )
=> ( r_is @ ( r_mn @ X0 @ X1 ) @ X2 ) ) ) ) )).

thf(satz187f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ ( r_pl @ X2 @ X1 ) @ X0 )
=> ( r_is @ X2 @ ( r_mn @ X0 @ X1 ) ) ) ) ) )).

thf(satz187,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_one
@ ^ [X2: \$i] :
( r_is @ ( r_pl @ X1 @ X2 ) @ X0 ) ) ) )).

thf(satz188a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X2 ) )
=> ( r_more @ X0 @ X1 ) ) ) ) )).

thf(satz188b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X2 ) )
=> ( r_is @ X0 @ X1 ) ) ) ) )).

thf(satz188c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X2 ) )
=> ( r_less @ X0 @ X1 ) ) ) ) )).

thf(satz188d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X0 @ X1 )
=> ( r_more @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz188e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ X0 @ X1 )
=> ( r_is @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz188f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( r_less @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz188g,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X2 @ X1 ) )
=> ( r_more @ X0 @ X1 ) ) ) ) )).

thf(satz188h,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X2 @ X1 ) )
=> ( r_is @ X0 @ X1 ) ) ) ) )).

thf(satz188j,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X2 @ X1 ) )
=> ( r_less @ X0 @ X1 ) ) ) ) )).

thf(satz188k,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X0 @ X1 )
=> ( r_more @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz188l,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ X0 @ X1 )
=> ( r_is @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz188m,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( r_less @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz188n,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_is @ X0 @ X1 )
=> ( ( r_more @ X2 @ X3 )
=> ( r_more @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz188o,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_is @ X0 @ X1 )
=> ( ( r_more @ X2 @ X3 )
=> ( r_more @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz188p,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_is @ X0 @ X1 )
=> ( ( r_less @ X2 @ X3 )
=> ( r_less @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz188q,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_is @ X0 @ X1 )
=> ( ( r_less @ X2 @ X3 )
=> ( r_less @ ( r_pl @ X2 @ X0 ) @ ( r_pl @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz189,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_more @ X0 @ X1 )
=> ( ( r_more @ X2 @ X3 )
=> ( r_more @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz189a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( r_less @ X2 @ X3 )
=> ( r_less @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz190a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_moreis @ X0 @ X1 )
=> ( ( r_more @ X2 @ X3 )
=> ( r_more @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz190b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_more @ X0 @ X1 )
=> ( ( r_moreis @ X2 @ X3 )
=> ( r_more @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz190c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_lessis @ X0 @ X1 )
=> ( ( r_less @ X2 @ X3 )
=> ( r_less @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz190d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( r_lessis @ X2 @ X3 )
=> ( r_less @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz191,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_moreis @ X0 @ X1 )
=> ( ( r_moreis @ X2 @ X3 )
=> ( r_moreis @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz191a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_lessis @ X0 @ X1 )
=> ( ( r_lessis @ X2 @ X3 )
=> ( r_lessis @ ( r_pl @ X0 @ X2 ) @ ( r_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(typ_timesdr,type,(
timesdr: \$i )).

thf(def_timesdr,definition,
( timesdr
= ( d_Sigma @ dif
@ ^ [X0: \$i] :
( d_Sigma @ dif
@ ^ [X1: \$i] :
( realof @ ( rp_td @ X0 @ X1 ) ) ) ) )).

thf(typ_r_ts,type,(
r_ts: \$i > \$i > \$i )).

thf(def_r_ts,definition,
( r_ts
= ( indreal2 @ real @ timesdr ) )).

thf(satz192a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ X0 @ r_0 )
=> ( r_is @ ( r_ts @ X0 @ X1 ) @ r_0 ) ) ) )).

thf(satz192b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ X1 @ r_0 )
=> ( r_is @ ( r_ts @ X0 @ X1 ) @ r_0 ) ) ) )).

thf(satz192c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_is @ ( r_ts @ X0 @ X1 ) @ r_0 )
=> ( l_or @ ( r_is @ X0 @ r_0 ) @ ( r_is @ X1 @ r_0 ) ) ) ) )).

thf(satz192d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X0 @ r_0 )
=> ( ( r_nis @ X1 @ r_0 )
=> ( r_nis @ ( r_ts @ X0 @ X1 ) @ r_0 ) ) ) ) )).

thf(satz193,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( abs @ ( r_ts @ X0 @ X1 ) ) @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) ) ) )).

thf(satz193a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) @ ( abs @ ( r_ts @ X0 @ X1 ) ) ) ) )).

thf(satz194,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_ts @ X1 @ X0 ) ) ) )).

thf(typ_d_1rl,type,(
d_1rl: \$i )).

thf(def_d_1rl,definition,
( d_1rl
= ( realof @ d_1df ) )).

thf(satz195,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ ( r_ts @ X0 @ d_1rl ) @ X0 ) )).

thf(satz195a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ X0 @ ( r_ts @ X0 @ d_1rl ) ) )).

thf(satz195b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ ( r_ts @ d_1rl @ X0 ) @ X0 ) )).

thf(satz195c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( r_is @ X0 @ ( r_ts @ d_1rl @ X0 ) ) )).

thf(satz196a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( pos @ X0 )
=> ( ( pos @ X1 )
=> ( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) ) ) ) ) )).

thf(satz196b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( neg @ X0 )
=> ( ( neg @ X1 )
=> ( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) ) ) ) ) )).

thf(satz196c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( pos @ X0 )
=> ( ( neg @ X1 )
=> ( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_m0 @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) ) ) ) ) ) )).

thf(satz196d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( neg @ X0 )
=> ( ( pos @ X1 )
=> ( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_m0 @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) ) ) ) ) ) )).

thf(satz196e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( d_not @ ( r_is @ X0 @ r_0 ) )
=> ( ( d_not @ ( r_is @ X1 @ r_0 ) )
=> ( ( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) )
=> ( l_or @ ( d_and @ ( pos @ X0 ) @ ( pos @ X1 ) ) @ ( d_and @ ( neg @ X0 ) @ ( neg @ X1 ) ) ) ) ) ) ) )).

thf(satz196f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( d_not @ ( r_is @ X0 @ r_0 ) )
=> ( ( d_not @ ( r_is @ X1 @ r_0 ) )
=> ( ( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_m0 @ ( r_ts @ ( abs @ X0 ) @ ( abs @ X1 ) ) ) )
=> ( l_or @ ( d_and @ ( pos @ X0 ) @ ( neg @ X1 ) ) @ ( d_and @ ( neg @ X0 ) @ ( pos @ X1 ) ) ) ) ) ) ) )).

thf(satz196g,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( pos @ ( r_ts @ X0 @ X1 ) )
=> ( l_or @ ( d_and @ ( pos @ X0 ) @ ( pos @ X1 ) ) @ ( d_and @ ( neg @ X0 ) @ ( neg @ X1 ) ) ) ) ) )).

thf(satz196h,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( neg @ ( r_ts @ X0 @ X1 ) )
=> ( l_or @ ( d_and @ ( pos @ X0 ) @ ( neg @ X1 ) ) @ ( d_and @ ( neg @ X0 ) @ ( pos @ X1 ) ) ) ) ) )).

thf(satz197a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ ( r_m0 @ X0 ) @ X1 ) @ ( r_m0 @ ( r_ts @ X0 @ X1 ) ) ) ) )).

thf(satz197b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ X0 @ ( r_m0 @ X1 ) ) @ ( r_m0 @ ( r_ts @ X0 @ X1 ) ) ) ) )).

thf(satz197c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ ( r_m0 @ X0 ) @ X1 ) @ ( r_ts @ X0 @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz197d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ X0 @ ( r_m0 @ X1 ) ) @ ( r_ts @ ( r_m0 @ X0 ) @ X1 ) ) ) )).

thf(satz197e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_m0 @ ( r_ts @ X0 @ X1 ) ) @ ( r_ts @ ( r_m0 @ X0 ) @ X1 ) ) ) )).

thf(satz197f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_m0 @ ( r_ts @ X0 @ X1 ) ) @ ( r_ts @ X0 @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz198,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) ) @ ( r_ts @ X0 @ X1 ) ) ) )).

thf(satz198a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( r_is @ ( r_ts @ X0 @ X1 ) @ ( r_ts @ ( r_m0 @ X0 ) @ ( r_m0 @ X1 ) ) ) ) )).

thf(satz199,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( r_is @ ( r_ts @ ( r_ts @ X0 @ X1 ) @ X2 ) @ ( r_ts @ X0 @ ( r_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz201,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( r_is @ ( r_ts @ X0 @ ( r_pl @ X1 @ X2 ) ) @ ( r_pl @ ( r_ts @ X0 @ X1 ) @ ( r_ts @ X0 @ X2 ) ) ) ) ) )).

thf(satz202,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( r_is @ ( r_ts @ X0 @ ( r_mn @ X1 @ X2 ) ) @ ( r_mn @ ( r_ts @ X0 @ X1 ) @ ( r_ts @ X0 @ X2 ) ) ) ) ) )).

thf(satz203a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X0 @ X1 )
=> ( ( pos @ X2 )
=> ( r_more @ ( r_ts @ X0 @ X2 ) @ ( r_ts @ X1 @ X2 ) ) ) ) ) ) )).

thf(satz203b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ X2 @ r_0 )
=> ( r_is @ ( r_ts @ X0 @ X2 ) @ ( r_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz203c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X0 @ X1 )
=> ( ( neg @ X2 )
=> ( r_less @ ( r_ts @ X0 @ X2 ) @ ( r_ts @ X1 @ X2 ) ) ) ) ) ) )).

thf(satz203d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X0 @ X1 )
=> ( ( pos @ X2 )
=> ( r_more @ ( r_ts @ X2 @ X0 ) @ ( r_ts @ X2 @ X1 ) ) ) ) ) ) )).

thf(satz203e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_is @ X2 @ r_0 )
=> ( r_is @ ( r_ts @ X2 @ X0 ) @ ( r_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz203f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X0 @ X1 )
=> ( ( neg @ X2 )
=> ( r_less @ ( r_ts @ X2 @ X0 ) @ ( r_ts @ X2 @ X1 ) ) ) ) ) ) )).

thf(satz203g,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( pos @ X2 )
=> ( r_less @ ( r_ts @ X0 @ X2 ) @ ( r_ts @ X1 @ X2 ) ) ) ) ) ) )).

thf(satz203j,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( neg @ X2 )
=> ( r_more @ ( r_ts @ X0 @ X2 ) @ ( r_ts @ X1 @ X2 ) ) ) ) ) ) )).

thf(satz203k,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( pos @ X2 )
=> ( r_less @ ( r_ts @ X2 @ X0 ) @ ( r_ts @ X2 @ X1 ) ) ) ) ) ) )).

thf(satz203m,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X0 @ X1 )
=> ( ( neg @ X2 )
=> ( r_more @ ( r_ts @ X2 @ X0 ) @ ( r_ts @ X2 @ X1 ) ) ) ) ) ) )).

thf(satz204b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( r_is @ ( r_ts @ X1 @ X2 ) @ X0 )
=> ( ( r_is @ ( r_ts @ X1 @ X3 ) @ X0 )
=> ( r_is @ X2 @ X3 ) ) ) ) ) ) ) )).

thf(satz204a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( r_some
@ ^ [X2: \$i] :
( r_is @ ( r_ts @ X1 @ X2 ) @ X0 ) ) ) ) )).

thf(satz204,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( r_one
@ ^ [X2: \$i] :
( r_is @ ( r_ts @ X1 @ X2 ) @ X0 ) ) ) ) )).

thf(typ_r_ov,type,(
r_ov: \$i > \$i > \$i )).

thf(def_r_ov,definition,
( r_ov
= ( ^ [X0: \$i,X1: \$i] :
( ind @ real
@ ^ [X2: \$i] :
( r_is @ ( r_ts @ X1 @ X2 ) @ X0 ) ) ) )).

thf(satz204c,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( r_is @ ( r_ts @ X1 @ ( r_ov @ X0 @ X1 ) ) @ X0 ) ) ) )).

thf(satz204d,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( r_is @ X0 @ ( r_ts @ X1 @ ( r_ov @ X0 @ X1 ) ) ) ) ) )).

thf(satz204e,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( r_is @ ( r_ts @ ( r_ov @ X0 @ X1 ) @ X1 ) @ X0 ) ) ) )).

thf(satz204f,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( r_is @ X0 @ ( r_ts @ ( r_ov @ X0 @ X1 ) @ X1 ) ) ) ) )).

thf(satz204g,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ real )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ real )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_nis @ X1 @ r_0 )
=> ( ( r_is @ ( r_ts @ X1 @ X2 ) @ X0 )
=> ( r_is @ X2 @ ( r_ov @ X0 @ X1 ) ) ) ) ) ) )).

thf(typ_s01,type,(
s01: \$i > \$i )).

thf(def_s01,definition,
( s01
= ( ^ [X0: \$i] :
( d_Sep @ real
@ ^ [X1: \$i] :
( r_lessis @ X1 @ X0 ) ) ) )).

thf(typ_s02,type,(
s02: \$i > \$i )).

thf(def_s02,definition,
( s02
= ( ^ [X0: \$i] :
( d_Sep @ real
@ ^ [X1: \$i] :
( r_more @ X1 @ X0 ) ) ) )).

thf(typ_s11,type,(
s11: \$i > \$i )).

thf(def_s11,definition,
( s11
= ( ^ [X0: \$i] :
( d_Sep @ real
@ ^ [X1: \$i] :
( r_less @ X1 @ X0 ) ) ) )).

thf(typ_s12,type,(
s12: \$i > \$i )).

thf(def_s12,definition,
( s12
= ( ^ [X0: \$i] :
( d_Sep @ real
@ ^ [X1: \$i] :
( r_moreis @ X1 @ X0 ) ) ) )).

thf(typ_d_2rl,type,(
d_2rl: \$i )).

thf(def_d_2rl,definition,
( d_2rl
= ( r_pl @ d_1rl @ d_1rl ) )).

thf(typ_half,type,(
half: \$i )).

thf(def_half,definition,
( half
= ( r_ov @ d_1rl @ d_2rl ) )).

thf(typ_d_5r205_prop1,type,(
d_5r205_prop1: \$i > \$i > \$o )).

thf(def_d_5r205_prop1,definition,
( d_5r205_prop1
= ( ^ [X0: \$i,X1: \$i] :
( r_all
@ ^ [X2: \$i] :
( ( r_less @ X2 @ X1 )
=> ( r_in @ X2 @ X0 ) ) ) ) )).

thf(typ_d_5r205_prop2,type,(
d_5r205_prop2: \$i > \$i > \$o )).

thf(def_d_5r205_prop2,definition,
( d_5r205_prop2
= ( ^ [X0: \$i,X1: \$i] :
( r_all
@ ^ [X2: \$i] :
( ( r_more @ X2 @ X1 )
=> ( r_in @ X2 @ X0 ) ) ) ) )).

thf(typ_d_5r205_prop3,type,(
d_5r205_prop3: \$i > \$i > \$i > \$o )).

thf(def_d_5r205_prop3,definition,
( d_5r205_prop3
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( d_and @ ( d_5r205_prop1 @ X0 @ X2 ) @ ( d_5r205_prop2 @ X1 @ X2 ) ) ) )).

thf(typ_mxy,type,(
mxy: \$i > \$i > \$i )).

thf(def_mxy,definition,
( mxy
= ( ^ [X0: \$i,X1: \$i] :
( r_ts @ half @ ( r_pl @ X0 @ X1 ) ) ) )).

thf(typ_sc1,type,(
sc1: \$i > \$i )).

thf(def_sc1,definition,
( sc1
= ( ^ [X0: \$i] :
( d_Sep @ cut
@ ^ [X1: \$i] :
( r_in @ ( pofrp @ X1 ) @ X0 ) ) ) )).

thf(typ_pr1,type,(
pr1: \$i > \$i > \$i )).

thf(def_pr1,definition,
( pr1
= ( ^ [X0: \$i] : rpofp ) )).

thf(typ_ps1,type,(
ps1: \$i > \$i > \$i > \$i > \$i )).

thf(def_ps1,definition,
( ps1
= ( ^ [X0: \$i,X1: \$i,X2: \$i] : rpofp ) )).

thf(typ_stc,type,(
stc: \$i > \$i > \$i > \$i )).

thf(def_stc,definition,
( stc
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( schnitt @ ( sc1 @ X0 ) @ ( sc1 @ X1 ) ) ) )).

thf(typ_stp,type,(
stp: \$i > \$i > \$i > \$i )).

thf(def_stp,definition,
( stp
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( pofrp @ ( stc @ X0 @ X1 @ X2 ) ) ) )).

thf(typ_d_5r205_sp1,type,(
d_5r205_sp1: \$i > \$i )).

thf(def_d_5r205_sp1,definition,
( d_5r205_sp1
= ( ^ [X0: \$i] :
( d_Sep @ real
@ ^ [X1: \$i] :
( r_in @ ( r_m0 @ X1 ) @ X0 ) ) ) )).

thf(satz205,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ ( power @ real ) )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ ( power @ real ) )
@ ^ [X1: \$i] :
( ( r_all
@ ^ [X2: \$i] :
( l_or @ ( r_in @ X2 @ X0 ) @ ( r_in @ X2 @ X1 ) ) )
=> ( ( nonempty @ real @ X0 )
=> ( ( nonempty @ real @ X1 )
=> ( ( r_all
@ ^ [X2: \$i] :
( ( r_in @ X2 @ X0 )
=> ( r_all
@ ^ [X3: \$i] :
( ( r_in @ X3 @ X1 )
=> ( r_less @ X2 @ X3 ) ) ) ) )
=> ( r_one
@ ^ [X2: \$i] :
( d_and
@ ( r_all
@ ^ [X3: \$i] :
( ( r_less @ X3 @ X2 )
=> ( r_in @ X3 @ X0 ) ) )
@ ( r_all
@ ^ [X3: \$i] :
( ( r_more @ X3 @ X2 )
=> ( r_in @ X3 @ X1 ) ) ) ) ) ) ) ) ) ) )).

thf(typ_r_schnitt,type,(
r_schnitt: \$i > \$i > \$i )).

thf(def_r_schnitt,definition,
( r_schnitt
= ( ^ [X0: \$i,X1: \$i] :
( ind @ real @ ( d_5r205_prop3 @ X0 @ X1 ) ) ) )).

thf(satz205a,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ ( power @ real ) )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ ( power @ real ) )
@ ^ [X1: \$i] :
( ( r_all
@ ^ [X2: \$i] :
( l_or @ ( r_in @ X2 @ X0 ) @ ( r_in @ X2 @ X1 ) ) )
=> ( ( nonempty @ real @ X0 )
=> ( ( nonempty @ real @ X1 )
=> ( ( r_all
@ ^ [X2: \$i] :
( ( r_in @ X2 @ X0 )
=> ( r_all
@ ^ [X3: \$i] :
( ( r_in @ X3 @ X1 )
=> ( r_less @ X2 @ X3 ) ) ) ) )
=> ( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_less @ X2 @ ( r_schnitt @ X0 @ X1 ) )
=> ( r_in @ X2 @ X0 ) ) ) ) ) ) ) ) )).

thf(satz205b,axiom,
( all_of
@ ^ [X0: \$i] :
( in @ X0 @ ( power @ real ) )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] :
( in @ X1 @ ( power @ real ) )
@ ^ [X1: \$i] :
( ( r_all
@ ^ [X2: \$i] :
( l_or @ ( r_in @ X2 @ X0 ) @ ( r_in @ X2 @ X1 ) ) )
=> ( ( nonempty @ real @ X0 )
=> ( ( nonempty @ real @ X1 )
=> ( ( r_all
@ ^ [X2: \$i] :
( ( r_in @ X2 @ X0 )
=> ( r_all
@ ^ [X3: \$i] :
( ( r_in @ X3 @ X1 )
=> ( r_less @ X2 @ X3 ) ) ) ) )
=> ( all_of
@ ^ [X2: \$i] :
( in @ X2 @ real )
@ ^ [X2: \$i] :
( ( r_more @ X2 @ ( r_schnitt @ X0 @ X1 ) )
=> ( r_in @ X2 @ X1 ) ) ) ) ) ) ) ) )).

thf(typ_r_sqrt,type,(
r_sqrt: \$i > \$i )).

thf(def_r_sqrt,definition,
( r_sqrt
= ( ^ [X0: \$i] :
( ind @ real
@ ^ [X1: \$i] :
( d_and @ ( d_not @ ( neg @ X1 ) ) @ ( r_is @ ( r_ts @ X1 @ X1 ) @ X0 ) ) ) ) )).

thf(typ_shiftl,type,(
shiftl: \$i > \$i > \$i )).

thf(def_shiftl,definition,
( shiftl
= ( ^ [X0: \$i,X1: \$i] :
( ntofrl @ ( r_mn @ ( r_pl @ X0 @ d_1rl ) @ X1 ) ) ) )).

thf(typ_shift_n1,type,(
shift_n1: \$i > \$i > \$i > \$i )).

thf(def_shift_n1,definition,
( shift_n1
= ( ^ [X0: \$i,X1: \$i] :
( inn @ ( shiftl @ X0 @ X1 ) ) ) )).

thf(typ_shift_n2,type,(
shift_n2: \$i > \$i > \$i > \$i )).

thf(def_shift_n2,definition,
( shift_n2
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( rlofnt @ ( shift_n1 @ X0 @ X1 @ X2 ) ) ) )).

thf(typ_shiftr,type,(
shiftr: \$i > \$i > \$i > \$i )).

thf(def_shiftr,definition,
( shiftr
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( r_mn @ ( r_pl @ ( shift_n2 @ X0 @ X1 @ X2 ) @ X1 ) @ d_1rl ) ) )).

thf(typ_shift_ul,type,(
shift_ul: \$i > \$i > \$i > \$i )).

thf(def_shift_ul,definition,
( shift_ul
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( shiftl @ X2 @ X1 ) ) )).

thf(typ_shiftl1,type,(
shiftl1: \$i > \$i > \$i > \$i )).

thf(def_shiftl1,definition,
( shiftl1
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( outn @ ( shiftl @ X0 @ X1 ) @ ( shift_ul @ X0 @ X1 @ X2 ) ) ) )).

thf(typ_seq,type,(
seq: \$i > \$i > \$i > \$i )).

thf(def_seq,definition,
( seq
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( d_Pi @ real
@ ^ [X3: \$i] :
( if @ ( intrl @ X3 ) @ ( if @ ( r_lessis @ X1 @ X3 ) @ ( if @ ( r_lessis @ X3 @ X0 ) @ X2 @ ( ordsucc @ emptyset ) ) @ ( ordsucc @ emptyset ) ) @ ( ordsucc @ emptyset ) ) ) ) )).

thf(typ_proofsirrelevant,type,(
proofsirrelevant: \$i > \$i > \$i > \$i > \$o )).

thf(def_proofsirrelevant,definition,
( proofsirrelevant
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] :
( all_of
@ ^ [X4: \$i] :
( in @ X4 @ real )
@ ^ [X4: \$i] :
( ( intrl @ X4 )
=> ( ( r_lessis @ X1 @ X4 )
=> ( ( r_lessis @ X4 @ X0 )
=> ( all_of
@ ^ [X5: \$i] :
( in @ X5 @ real )
@ ^ [X5: \$i] :
( ( intrl @ X5 )
=> ( ( r_lessis @ X1 @ X5 )
=> ( ( r_lessis @ X5 @ X0 )
=> ( ( r_is @ X4 @ X5 )
=> ( e_is @ X2 @ ( ap @ X3 @ X4 ) @ ( ap @ X3 @ X5 ) ) ) ) ) ) ) ) ) ) ) ) )).

thf(typ_shiftf,type,(
shiftf: \$i > \$i > \$i > \$i > \$i )).

thf(def_shiftf,definition,
( shiftf
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] :
( d_Sigma @ ( d_1to @ ( shiftl @ X0 @ X1 ) )
@ ^ [X4: \$i] :
( ap @ X3 @ ( shiftr @ X0 @ X1 @ X4 ) ) ) ) )).

thf(typ_inseq,type,(
inseq: \$i > \$i > \$i > \$o )).

thf(def_inseq,definition,
( inseq
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( intrl @ X3 )
=> ( ( r_lessis @ X1 @ X3 )
=> ( ( r_lessis @ X3 @ X0 )
=> ( and3 @ ( intrl @ ( ap @ X2 @ X3 ) ) @ ( r_lessis @ X1 @ ( ap @ X2 @ X3 ) ) @ ( r_lessis @ ( ap @ X2 @ X3 ) @ X0 ) ) ) ) ) ) ) )).

thf(typ_injseq,type,(
injseq: \$i > \$i > \$i > \$o )).

thf(def_injseq,definition,
( injseq
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( intrl @ X3 )
=> ( ( r_lessis @ X1 @ X3 )
=> ( ( r_lessis @ X3 @ X0 )
=> ( all_of
@ ^ [X4: \$i] :
( in @ X4 @ real )
@ ^ [X4: \$i] :
( ( intrl @ X4 )
=> ( ( r_lessis @ X1 @ X4 )
=> ( ( r_lessis @ X4 @ X0 )
=> ( ( r_is @ ( ap @ X2 @ X3 ) @ ( ap @ X2 @ X4 ) )
=> ( r_is @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) )).

thf(typ_shift_prop1,type,(
shift_prop1: \$i > \$i > \$i > \$i > \$i > \$o )).

thf(def_shift_prop1,definition,
( shift_prop1
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i,X4: \$i] :
( r_is @ X3 @ ( ap @ X2 @ X4 ) ) ) )).

thf(typ_improp,type,(
improp: \$i > \$i > \$i > \$i > \$i > \$o )).

thf(def_improp,definition,
( improp
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i,X4: \$i] :
( d_and @ ( and3 @ ( intrl @ X4 ) @ ( r_lessis @ X1 @ X4 ) @ ( r_lessis @ X4 @ X0 ) )
@ ( ( and3 @ ( intrl @ X4 ) @ ( r_lessis @ X1 @ X4 ) @ ( r_lessis @ X4 @ X0 ) )
=> ( shift_prop1 @ X0 @ X1 @ X2 @ X3 @ X4 ) ) ) ) )).

thf(typ_imseq,type,(
imseq: \$i > \$i > \$i > \$i > \$o )).

thf(def_imseq,definition,
( imseq
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] :
( r_some @ ( improp @ X0 @ X1 @ X2 @ X3 ) ) ) )).

thf(typ_surjseq,type,(
surjseq: \$i > \$i > \$i > \$o )).

thf(def_surjseq,definition,
( surjseq
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( all_of
@ ^ [X3: \$i] :
( in @ X3 @ real )
@ ^ [X3: \$i] :
( ( intrl @ X3 )
=> ( ( r_lessis @ X1 @ X3 )
=> ( ( r_lessis @ X3 @ X0 )
=> ( imseq @ X0 @ X1 @ X2 @ X3 ) ) ) ) ) ) )).

thf(typ_perm,type,(
perm: \$i > \$i > \$i > \$o )).

thf(def_perm,definition,
( perm
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( d_and @ ( injseq @ X0 @ X1 @ X2 ) @ ( surjseq @ X0 @ X1 @ X2 ) ) ) )).

thf(typ_shift_ns,type,(
shift_ns: \$i > \$i > \$i > \$i > \$i )).

thf(def_shift_ns,definition,
( shift_ns
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] :
( ap @ X2 @ ( shiftr @ X0 @ X1 @ X3 ) ) ) )).

thf(typ_shiftseq,type,(
shiftseq: \$i > \$i > \$i > \$i )).

thf(def_shiftseq,definition,
( shiftseq
= ( ^ [X0: \$i,X1: \$i,X2: \$i] :
( d_Sigma @ ( d_1to @ ( shiftl @ X0 @ X1 ) )
@ ^ [X3: \$i] :
( shiftl1 @ X0 @ X1 @ ( shift_ns @ X0 @ X1 @ X2 @ X3 ) ) ) ) )).

thf(typ_ul1,type,(
ul1: \$i > \$i > \$i > \$i > \$i > \$i )).

thf(def_ul1,definition,
( ul1
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] :
( shiftl1 @ X0 @ X1 ) ) )).

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