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      :

% Comments :
%------------------------------------------------------------------------------
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 ) ) )).

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