TPTP Problem File: NUM690^4.p
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%------------------------------------------------------------------------------
% File : NUM690^4 : TPTP v9.0.0. Released v7.1.0.
% Domain : Number theory
% Problem : Grundlagen problem satz22d
% Version : [Bro17] axioms : Especial.
% English :
% Refs : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source : [Br017]
% Names :
% Status : Theorem
% Rating : 0.62 v9.0.0, 0.80 v8.2.0, 0.92 v8.1.0, 0.91 v7.5.0, 0.86 v7.4.0, 0.89 v7.2.0, 0.88 v7.1.0
% Syntax : Number of formulae : 394 ( 136 unt; 131 typ; 124 def)
% Number of atoms : 1447 ( 154 equ; 0 cnn)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 2659 ( 7 ~; 4 |; 14 &;2487 @)
% ( 3 <=>; 144 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 443 ( 443 >; 0 *; 0 +; 0 <<)
% Number of symbols : 156 ( 154 usr; 29 con; 0-7 aty)
% Number of variables : 916 ( 759 ^; 149 !; 8 ?; 916 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
%------------------------------------------------------------------------------
include('Axioms/NUM007^0.ax').
%------------------------------------------------------------------------------
thf(satz1,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( nis @ X0 @ X1 )
=> ( nis @ ( ordsucc @ X0 ) @ ( ordsucc @ X1 ) ) ) ) ) ).
thf(typ_d_22_prop1,type,
d_22_prop1: $i > $o ).
thf(def_d_22_prop1,definition,
( d_22_prop1
= ( ^ [X0: $i] : ( nis @ ( ordsucc @ X0 ) @ X0 ) ) ) ).
thf(satz2,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( nis @ ( ordsucc @ X0 ) @ X0 ) ) ).
thf(typ_d_23_prop1,type,
d_23_prop1: $i > $o ).
thf(def_d_23_prop1,definition,
( d_23_prop1
= ( ^ [X0: $i] :
( l_or @ ( n_is @ X0 @ n_1 )
@ ( n_some
@ ^ [X1: $i] : ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ) ).
thf(satz3,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( ( nis @ X0 @ n_1 )
=> ( n_some
@ ^ [X1: $i] : ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ).
thf(satz3a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( ( nis @ X0 @ n_1 )
=> ( n_one
@ ^ [X1: $i] : ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ).
thf(typ_d_24_prop1,type,
d_24_prop1: $i > $o ).
thf(def_d_24_prop1,definition,
( d_24_prop1
= ( ^ [X0: $i] :
( n_all
@ ^ [X1: $i] : ( n_is @ ( ap @ X0 @ ( ordsucc @ X1 ) ) @ ( ordsucc @ ( ap @ X0 @ X1 ) ) ) ) ) ) ).
thf(typ_d_24_prop2,type,
d_24_prop2: $i > $i > $o ).
thf(def_d_24_prop2,definition,
( d_24_prop2
= ( ^ [X0: $i,X1: $i] : ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ ( ordsucc @ X0 ) ) @ ( d_24_prop1 @ X1 ) ) ) ) ).
thf(typ_prop3,type,
prop3: $i > $i > $i > $o ).
thf(def_prop3,definition,
( prop3
= ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ ( ap @ X0 @ X2 ) @ ( ap @ X1 @ X2 ) ) ) ) ).
thf(typ_prop4,type,
prop4: $i > $o ).
thf(def_prop4,definition,
( prop4
= ( ^ [X0: $i] :
( l_some
@ ( d_Pi @ nat
@ ^ [X1: $i] : nat )
@ ( d_24_prop2 @ X0 ) ) ) ) ).
thf(typ_d_24_g,type,
d_24_g: $i > $i ).
thf(def_d_24_g,definition,
( d_24_g
= ( ^ [X0: $i] :
( d_Sigma @ nat
@ ^ [X1: $i] : ( ordsucc @ ( ap @ X0 @ X1 ) ) ) ) ) ).
thf(satz4,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( one
@ ( d_Pi @ nat
@ ^ [X1: $i] : nat )
@ ^ [X1: $i] :
( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ ( ordsucc @ X0 ) )
@ ( n_all
@ ^ [X2: $i] : ( n_is @ ( ap @ X1 @ ( ordsucc @ X2 ) ) @ ( ordsucc @ ( ap @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(typ_plus,type,
plus: $i > $i ).
thf(def_plus,definition,
( plus
= ( ^ [X0: $i] :
( ind
@ ( d_Pi @ nat
@ ^ [X1: $i] : nat )
@ ( d_24_prop2 @ X0 ) ) ) ) ).
thf(typ_n_pl,type,
n_pl: $i > $i > $i ).
thf(def_n_pl,definition,
( n_pl
= ( ^ [X0: $i] : ( ap @ ( plus @ X0 ) ) ) ) ).
thf(satz4a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ ( n_pl @ X0 @ n_1 ) @ ( ordsucc @ X0 ) ) ) ).
thf(satz4b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_pl @ X0 @ ( ordsucc @ X1 ) ) @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) ) ) ) ).
thf(satz4c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ ( n_pl @ n_1 @ X0 ) @ ( ordsucc @ X0 ) ) ) ).
thf(satz4d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_pl @ ( ordsucc @ X0 ) @ X1 ) @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) ) ) ) ).
thf(satz4e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ ( ordsucc @ X0 ) @ ( n_pl @ X0 @ n_1 ) ) ) ).
thf(satz4f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) @ ( n_pl @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ).
thf(satz4g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ ( ordsucc @ X0 ) @ ( n_pl @ n_1 @ X0 ) ) ) ).
thf(satz4h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) @ ( n_pl @ ( ordsucc @ X0 ) @ X1 ) ) ) ) ).
thf(typ_d_25_prop1,type,
d_25_prop1: $i > $i > $i > $o ).
thf(def_d_25_prop1,definition,
( d_25_prop1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ ( n_pl @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pl @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ) ).
thf(satz5,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] : ( n_is @ ( n_pl @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pl @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(typ_d_26_prop1,type,
d_26_prop1: $i > $i > $o ).
thf(def_d_26_prop1,definition,
( d_26_prop1
= ( ^ [X0: $i,X1: $i] : ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X1 @ X0 ) ) ) ) ).
thf(satz6,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X1 @ X0 ) ) ) ) ).
thf(typ_d_27_prop1,type,
d_27_prop1: $i > $i > $o ).
thf(def_d_27_prop1,definition,
( d_27_prop1
= ( ^ [X0: $i,X1: $i] : ( nis @ X1 @ ( n_pl @ X0 @ X1 ) ) ) ) ).
thf(satz7,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( nis @ X1 @ ( n_pl @ X0 @ X1 ) ) ) ) ).
thf(typ_d_28_prop1,type,
d_28_prop1: $i > $i > $i > $o ).
thf(def_d_28_prop1,definition,
( d_28_prop1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( nis @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) ) ) ) ).
thf(satz8,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( nis @ X1 @ X2 )
=> ( nis @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz8a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) )
=> ( n_is @ X1 @ X2 ) ) ) ) ) ).
thf(typ_diffprop,type,
diffprop: $i > $i > $i > $o ).
thf(def_diffprop,definition,
( diffprop
= ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ).
thf(satz8b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( amone @ nat
@ ^ [X2: $i] : ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ) ).
thf(typ_d_29_ii,type,
d_29_ii: $i > $i > $o ).
thf(def_d_29_ii,definition,
( d_29_ii
= ( ^ [X0: $i,X1: $i] : ( n_some @ ( diffprop @ X0 @ X1 ) ) ) ) ).
thf(typ_iii,type,
iii: $i > $i > $o ).
thf(def_iii,definition,
( iii
= ( ^ [X0: $i,X1: $i] : ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) ).
thf(typ_d_29_prop1,type,
d_29_prop1: $i > $i > $o ).
thf(def_d_29_prop1,definition,
( d_29_prop1
= ( ^ [X0: $i,X1: $i] : ( or3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).
thf(satz9,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( orec3 @ ( n_is @ X0 @ X1 )
@ ( n_some
@ ^ [X2: $i] : ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) )
@ ( n_some
@ ^ [X2: $i] : ( n_is @ X1 @ ( n_pl @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz9a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( or3 @ ( n_is @ X0 @ X1 ) @ ( n_some @ ( diffprop @ X0 @ X1 ) ) @ ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) ) ).
thf(satz9b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( ec3 @ ( n_is @ X0 @ X1 ) @ ( n_some @ ( diffprop @ X0 @ X1 ) ) @ ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) ) ).
thf(satz10,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( orec3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).
thf(satz10a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( or3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).
thf(satz10b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( ec3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).
thf(satz11,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_29_ii @ X0 @ X1 )
=> ( iii @ X1 @ X0 ) ) ) ) ).
thf(satz12,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( iii @ X0 @ X1 )
=> ( d_29_ii @ X1 @ X0 ) ) ) ) ).
thf(typ_moreis,type,
moreis: $i > $i > $o ).
thf(def_moreis,definition,
( moreis
= ( ^ [X0: $i,X1: $i] : ( l_or @ ( d_29_ii @ X0 @ X1 ) @ ( n_is @ X0 @ X1 ) ) ) ) ).
thf(typ_lessis,type,
lessis: $i > $i > $o ).
thf(def_lessis,definition,
( lessis
= ( ^ [X0: $i,X1: $i] : ( l_or @ ( iii @ X0 @ X1 ) @ ( n_is @ X0 @ X1 ) ) ) ) ).
thf(satz13,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( moreis @ X0 @ X1 )
=> ( lessis @ X1 @ X0 ) ) ) ) ).
thf(satz14,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( lessis @ X0 @ X1 )
=> ( moreis @ X1 @ X0 ) ) ) ) ).
thf(satz10c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( moreis @ X0 @ X1 )
=> ( d_not @ ( iii @ X0 @ X1 ) ) ) ) ) ).
thf(satz10d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( lessis @ X0 @ X1 )
=> ( d_not @ ( d_29_ii @ X0 @ X1 ) ) ) ) ) ).
thf(satz10e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_not @ ( d_29_ii @ X0 @ X1 ) )
=> ( lessis @ X0 @ X1 ) ) ) ) ).
thf(satz10f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_not @ ( iii @ X0 @ X1 ) )
=> ( moreis @ X0 @ X1 ) ) ) ) ).
thf(satz10g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_29_ii @ X0 @ X1 )
=> ( d_not @ ( lessis @ X0 @ X1 ) ) ) ) ) ).
thf(satz10h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( iii @ X0 @ X1 )
=> ( d_not @ ( moreis @ X0 @ X1 ) ) ) ) ) ).
thf(satz10j,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_not @ ( moreis @ X0 @ X1 ) )
=> ( iii @ X0 @ X1 ) ) ) ) ).
thf(satz10k,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_not @ ( lessis @ X0 @ X1 ) )
=> ( d_29_ii @ X0 @ X1 ) ) ) ) ).
thf(satz15,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( iii @ X0 @ X1 )
=> ( ( iii @ X1 @ X2 )
=> ( iii @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz16a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( lessis @ X0 @ X1 )
=> ( ( iii @ X1 @ X2 )
=> ( iii @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz16b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( iii @ X0 @ X1 )
=> ( ( lessis @ X1 @ X2 )
=> ( iii @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz16c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( moreis @ X0 @ X1 )
=> ( ( d_29_ii @ X1 @ X2 )
=> ( d_29_ii @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz16d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( d_29_ii @ X0 @ X1 )
=> ( ( moreis @ X1 @ X2 )
=> ( d_29_ii @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz17,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( lessis @ X0 @ X1 )
=> ( ( lessis @ X1 @ X2 )
=> ( lessis @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz18,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( d_29_ii @ ( n_pl @ X0 @ X1 ) @ X0 ) ) ) ).
thf(satz18a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( iii @ X0 @ ( n_pl @ X0 @ X1 ) ) ) ) ).
thf(satz18b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( d_29_ii @ ( ordsucc @ X0 ) @ X0 ) ) ).
thf(satz18c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( iii @ X0 @ ( ordsucc @ X0 ) ) ) ).
thf(satz19a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( d_29_ii @ X0 @ X1 )
=> ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz19b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( n_is @ X0 @ X1 )
=> ( n_is @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz19c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( iii @ X0 @ X1 )
=> ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz19d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( d_29_ii @ X0 @ X1 )
=> ( d_29_ii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz19e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( n_is @ X0 @ X1 )
=> ( n_is @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz19f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( iii @ X0 @ X1 )
=> ( iii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz19g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( n_is @ X0 @ X1 )
=> ( ( d_29_ii @ X2 @ X3 )
=> ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz19h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( n_is @ X0 @ X1 )
=> ( ( d_29_ii @ X2 @ X3 )
=> ( d_29_ii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz19j,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( n_is @ X0 @ X1 )
=> ( ( iii @ X2 @ X3 )
=> ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz19k,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( n_is @ X0 @ X1 )
=> ( ( iii @ X2 @ X3 )
=> ( iii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz19l,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( moreis @ X0 @ X1 )
=> ( moreis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz19m,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( moreis @ X0 @ X1 )
=> ( moreis @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz19n,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( lessis @ X0 @ X1 )
=> ( lessis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz19o,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( lessis @ X0 @ X1 )
=> ( lessis @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz20a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) )
=> ( d_29_ii @ X0 @ X1 ) ) ) ) ) ).
thf(satz20b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( n_is @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) )
=> ( n_is @ X0 @ X1 ) ) ) ) ) ).
thf(satz20c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) )
=> ( iii @ X0 @ X1 ) ) ) ) ) ).
thf(satz20d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( d_29_ii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) )
=> ( d_29_ii @ X0 @ X1 ) ) ) ) ) ).
thf(satz20e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( n_is @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) )
=> ( n_is @ X0 @ X1 ) ) ) ) ) ).
thf(satz20f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( ( iii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) )
=> ( iii @ X0 @ X1 ) ) ) ) ) ).
thf(satz21,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( d_29_ii @ X0 @ X1 )
=> ( ( d_29_ii @ X2 @ X3 )
=> ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz21a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( iii @ X0 @ X1 )
=> ( ( iii @ X2 @ X3 )
=> ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz22a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( moreis @ X0 @ X1 )
=> ( ( d_29_ii @ X2 @ X3 )
=> ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz22b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( d_29_ii @ X0 @ X1 )
=> ( ( moreis @ X2 @ X3 )
=> ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz22c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( lessis @ X0 @ X1 )
=> ( ( iii @ X2 @ X3 )
=> ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz22d,conjecture,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ nat )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ nat )
@ ^ [X3: $i] :
( ( iii @ X0 @ X1 )
=> ( ( lessis @ X2 @ X3 )
=> ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------