TPTP Problem File: NUM769^4.p
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%------------------------------------------------------------------------------
% File : NUM769^4 : TPTP v8.2.0. Released v7.1.0.
% Domain : Number theory
% Problem : Grundlagen problem satz67d
% Version : [Bro17] axioms : Especial.
% English :
% Refs : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source : [Br017]
% Names :
% Status : Theorem
% Rating : 0.80 v8.2.0, 0.77 v8.1.0, 0.82 v7.5.0, 0.86 v7.4.0, 0.67 v7.2.0, 0.62 v7.1.0
% Syntax : Number of formulae : 613 ( 183 unt; 179 typ; 172 def)
% Number of atoms : 2757 ( 202 equ; 0 cnn)
% Maximal formula atoms : 18 ( 6 avg)
% Number of connectives : 5312 ( 7 ~; 4 |; 14 &;5006 @)
% ( 3 <=>; 278 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 538 ( 538 >; 0 *; 0 +; 0 <<)
% Number of symbols : 210 ( 208 usr; 41 con; 0-7 aty)
% Number of variables : 1679 (1520 ^; 151 !; 8 ?;1679 :)
% 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,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 )
=> ( ( lessis @ X2 @ X3 )
=> ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz23,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 )
=> ( ( moreis @ X2 @ X3 )
=> ( moreis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz23a,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 )
=> ( ( lessis @ X2 @ X3 )
=> ( lessis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz24,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( moreis @ X0 @ n_1 ) ) ).
thf(satz24a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ( lessis @ n_1 ) ) ).
thf(satz24b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( d_29_ii @ ( ordsucc @ X0 ) @ n_1 ) ) ).
thf(satz24c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( iii @ n_1 @ ( ordsucc @ X0 ) ) ) ).
thf(satz25,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_29_ii @ X1 @ X0 )
=> ( moreis @ X1 @ ( n_pl @ X0 @ n_1 ) ) ) ) ) ).
thf(satz25a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_29_ii @ X1 @ X0 )
=> ( moreis @ X1 @ ( ordsucc @ X0 ) ) ) ) ) ).
thf(satz25b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( iii @ X1 @ X0 )
=> ( lessis @ ( n_pl @ X1 @ n_1 ) @ X0 ) ) ) ) ).
thf(satz25c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( iii @ X1 @ X0 )
=> ( lessis @ ( ordsucc @ X1 ) @ X0 ) ) ) ) ).
thf(satz26,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( iii @ X1 @ ( n_pl @ X0 @ n_1 ) )
=> ( lessis @ X1 @ X0 ) ) ) ) ).
thf(satz26a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( iii @ X1 @ ( ordsucc @ X0 ) )
=> ( lessis @ X1 @ X0 ) ) ) ) ).
thf(satz26b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_29_ii @ ( n_pl @ X1 @ n_1 ) @ X0 )
=> ( moreis @ X1 @ X0 ) ) ) ) ).
thf(satz26c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] :
( ( d_29_ii @ ( ordsucc @ X1 ) @ X0 )
=> ( moreis @ X1 @ X0 ) ) ) ) ).
thf(typ_lbprop,type,
lbprop: ( $i > $o ) > $i > $i > $o ).
thf(def_lbprop,definition,
( lbprop
= ( ^ [X0: $i > $o,X1: $i,X2: $i] : ( imp @ ( X0 @ X2 ) @ ( lessis @ X1 @ X2 ) ) ) ) ).
thf(typ_n_lb,type,
n_lb: ( $i > $o ) > $i > $o ).
thf(def_n_lb,definition,
( n_lb
= ( ^ [X0: $i > $o,X1: $i] : ( n_all @ ( lbprop @ X0 @ X1 ) ) ) ) ).
thf(typ_min,type,
min: ( $i > $o ) > $i > $o ).
thf(def_min,definition,
( min
= ( ^ [X0: $i > $o,X1: $i] : ( d_and @ ( n_lb @ X0 @ X1 ) @ ( X0 @ X1 ) ) ) ) ).
thf(satz27,axiom,
! [X0: $i > $o] :
( ( n_some @ X0 )
=> ( n_some @ ( min @ X0 ) ) ) ).
thf(satz27a,axiom,
! [X0: $i > $o] :
( ( n_some @ X0 )
=> ( n_one @ ( min @ X0 ) ) ) ).
thf(typ_d_428_prop1,type,
d_428_prop1: $i > $i > $o ).
thf(def_d_428_prop1,definition,
( d_428_prop1
= ( ^ [X0: $i,X1: $i] :
( n_all
@ ^ [X2: $i] : ( n_is @ ( ap @ X1 @ ( ordsucc @ X2 ) ) @ ( n_pl @ ( ap @ X1 @ X2 ) @ X0 ) ) ) ) ) ).
thf(typ_d_428_prop2,type,
d_428_prop2: $i > $i > $o ).
thf(def_d_428_prop2,definition,
( d_428_prop2
= ( ^ [X0: $i,X1: $i] : ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ X0 ) @ ( d_428_prop1 @ X0 @ X1 ) ) ) ) ).
thf(typ_d_428_prop4,type,
d_428_prop4: $i > $o ).
thf(def_d_428_prop4,definition,
( d_428_prop4
= ( ^ [X0: $i] :
( l_some
@ ( d_Pi @ nat
@ ^ [X1: $i] : nat )
@ ( d_428_prop2 @ X0 ) ) ) ) ).
thf(typ_d_428_id,type,
d_428_id: $i ).
thf(def_d_428_id,definition,
( d_428_id
= ( d_Sigma @ nat
@ ^ [X0: $i] : X0 ) ) ).
thf(typ_d_428_g,type,
d_428_g: $i > $i ).
thf(def_d_428_g,definition,
( d_428_g
= ( ^ [X0: $i] :
( d_Sigma @ nat
@ ^ [X1: $i] : ( n_pl @ ( ap @ X0 @ X1 ) @ X1 ) ) ) ) ).
thf(satz28,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 ) @ X0 )
@ ( n_all
@ ^ [X2: $i] : ( n_is @ ( ap @ X1 @ ( ordsucc @ X2 ) ) @ ( n_pl @ ( ap @ X1 @ X2 ) @ X0 ) ) ) ) ) ) ).
thf(typ_times,type,
times: $i > $i ).
thf(def_times,definition,
( times
= ( ^ [X0: $i] :
( ind
@ ( d_Pi @ nat
@ ^ [X1: $i] : nat )
@ ( d_428_prop2 @ X0 ) ) ) ) ).
thf(typ_n_ts,type,
n_ts: $i > $i > $i ).
thf(def_n_ts,definition,
( n_ts
= ( ^ [X0: $i] : ( ap @ ( times @ X0 ) ) ) ) ).
thf(satz28a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ ( n_ts @ X0 @ n_1 ) @ X0 ) ) ).
thf(satz28b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_ts @ X0 @ ( ordsucc @ X1 ) ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X0 ) ) ) ) ).
thf(satz28c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ ( n_ts @ n_1 @ X0 ) @ X0 ) ) ).
thf(satz28d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_ts @ ( ordsucc @ X0 ) @ X1 ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X1 ) ) ) ) ).
thf(satz28e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ X0 @ ( n_ts @ X0 @ n_1 ) ) ) ).
thf(satz28f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X0 ) @ ( n_ts @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ).
thf(satz28g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] : ( n_is @ X0 @ ( n_ts @ n_1 @ X0 ) ) ) ).
thf(satz28h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X1 ) @ ( n_ts @ ( ordsucc @ X0 ) @ X1 ) ) ) ) ).
thf(typ_d_429_prop1,type,
d_429_prop1: $i > $i > $o ).
thf(def_d_429_prop1,definition,
( d_429_prop1
= ( ^ [X0: $i,X1: $i] : ( n_is @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X1 @ X0 ) ) ) ) ).
thf(satz29,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ nat )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_is @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X1 @ X0 ) ) ) ) ).
thf(typ_d_430_prop1,type,
d_430_prop1: $i > $i > $i > $o ).
thf(def_d_430_prop1,definition,
( d_430_prop1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ ( n_ts @ X0 @ ( n_pl @ X1 @ X2 ) ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X0 @ X2 ) ) ) ) ) ).
thf(satz30,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_ts @ X0 @ ( n_pl @ X1 @ X2 ) ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X0 @ X2 ) ) ) ) ) ) ).
thf(typ_d_431_prop1,type,
d_431_prop1: $i > $i > $i > $o ).
thf(def_d_431_prop1,definition,
( d_431_prop1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ ( n_ts @ ( n_ts @ X0 @ X1 ) @ X2 ) @ ( n_ts @ X0 @ ( n_ts @ X1 @ X2 ) ) ) ) ) ).
thf(satz31,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_ts @ ( n_ts @ X0 @ X1 ) @ X2 ) @ ( n_ts @ X0 @ ( n_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz32a,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz32b,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz32c,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz32d,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_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz32e,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_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz32f,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_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz32g,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz32h,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_ts @ X2 @ X0 ) @ ( n_ts @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz32j,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz32k,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_ts @ X2 @ X0 ) @ ( n_ts @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz32l,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz32m,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_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz32n,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz32o,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_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz33a,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) )
=> ( d_29_ii @ X0 @ X1 ) ) ) ) ) ).
thf(satz33b,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) )
=> ( n_is @ X0 @ X1 ) ) ) ) ) ).
thf(satz33c,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) )
=> ( iii @ X0 @ X1 ) ) ) ) ) ).
thf(satz34,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz34a,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz35a,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz35b,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz35c,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_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz35d,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 )
=> ( ( lessis @ X2 @ X3 )
=> ( iii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz36,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 )
=> ( ( moreis @ X2 @ X3 )
=> ( moreis @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz36a,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 )
=> ( ( lessis @ X2 @ X3 )
=> ( lessis @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(typ_n_mn,type,
n_mn: $i > $i > $i ).
thf(def_n_mn,definition,
( n_mn
= ( ^ [X0: $i,X1: $i] : ( ind @ nat @ ( diffprop @ X0 @ X1 ) ) ) ) ).
thf(typ_d_1to,type,
d_1to: $i > $i ).
thf(def_d_1to,definition,
( d_1to
= ( ^ [X0: $i] :
( d_Sep @ nat
@ ^ [X1: $i] : ( lessis @ X1 @ X0 ) ) ) ) ).
thf(typ_outn,type,
outn: $i > $i > $i ).
thf(def_outn,definition,
( outn
= ( ^ [X0: $i] :
( out @ nat
@ ^ [X1: $i] : ( lessis @ X1 @ X0 ) ) ) ) ).
thf(typ_inn,type,
inn: $i > $i > $i ).
thf(def_inn,definition,
( inn
= ( ^ [X0: $i] :
( e_in @ nat
@ ^ [X1: $i] : ( lessis @ X1 @ X0 ) ) ) ) ).
thf(typ_n_1o,type,
n_1o: $i ).
thf(def_n_1o,definition,
( n_1o
= ( outn @ n_1 @ n_1 ) ) ).
thf(typ_singlet_u0,type,
singlet_u0: $i > $i ).
thf(def_singlet_u0,definition,
( singlet_u0
= ( inn @ n_1 ) ) ).
thf(typ_n_2,type,
n_2: $i ).
thf(def_n_2,definition,
( n_2
= ( n_pl @ n_1 @ n_1 ) ) ).
thf(typ_n_1t,type,
n_1t: $i ).
thf(def_n_1t,definition,
( n_1t
= ( outn @ n_2 @ n_1 ) ) ).
thf(typ_n_2t,type,
n_2t: $i ).
thf(def_n_2t,definition,
( n_2t
= ( outn @ n_2 @ n_2 ) ) ).
thf(typ_pair_u0,type,
pair_u0: $i > $i ).
thf(def_pair_u0,definition,
( pair_u0
= ( inn @ n_2 ) ) ).
thf(typ_pair1type,type,
pair1type: $i > $i ).
thf(def_pair1type,definition,
( pair1type
= ( ^ [X0: $i] :
( d_Pi @ ( d_1to @ n_2 )
@ ^ [X1: $i] : X0 ) ) ) ).
thf(typ_pair1,type,
pair1: $i > $i > $i > $i ).
thf(def_pair1,definition,
( pair1
= ( ^ [X0: $i,X1: $i,X2: $i] :
( d_Sigma @ ( d_1to @ n_2 )
@ ^ [X3: $i] : ( ite @ ( e_is @ ( d_1to @ n_2 ) @ X3 @ n_1t ) @ X0 @ X1 @ X2 ) ) ) ) ).
thf(typ_first1,type,
first1: $i > $i > $i ).
thf(def_first1,definition,
( first1
= ( ^ [X0: $i,X1: $i] : ( ap @ X1 @ n_1t ) ) ) ).
thf(typ_second1,type,
second1: $i > $i > $i ).
thf(def_second1,definition,
( second1
= ( ^ [X0: $i,X1: $i] : ( ap @ X1 @ n_2t ) ) ) ).
thf(typ_pair_q0,type,
pair_q0: $i > $i > $i ).
thf(def_pair_q0,definition,
( pair_q0
= ( ^ [X0: $i,X1: $i] : ( pair1 @ X0 @ ( first1 @ X0 @ X1 ) @ ( second1 @ X0 @ X1 ) ) ) ) ).
thf(typ_d_1out,type,
d_1out: $i > $i ).
thf(def_d_1out,definition,
( d_1out
= ( ^ [X0: $i] : ( outn @ X0 @ n_1 ) ) ) ).
thf(typ_xout,type,
xout: $i > $i ).
thf(def_xout,definition,
( xout
= ( ^ [X0: $i] : ( outn @ X0 @ X0 ) ) ) ).
thf(typ_left1to,type,
left1to: $i > $i > $i > $i ).
thf(def_left1to,definition,
( left1to
= ( ^ [X0: $i,X1: $i,X2: $i] : ( outn @ X0 @ ( inn @ X1 @ X2 ) ) ) ) ).
thf(typ_right1to,type,
right1to: $i > $i > $i > $i ).
thf(def_right1to,definition,
( right1to
= ( ^ [X0: $i,X1: $i,X2: $i] : ( outn @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ ( inn @ X1 @ X2 ) ) ) ) ) ).
thf(typ_left,type,
left: $i > $i > $i > $i > $i ).
thf(def_left,definition,
( left
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
( d_Sigma @ ( d_1to @ X2 )
@ ^ [X4: $i] : ( ap @ X3 @ ( left1to @ X1 @ X2 @ X4 ) ) ) ) ) ).
thf(typ_right,type,
right: $i > $i > $i > $i > $i ).
thf(def_right,definition,
( right
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
( d_Sigma @ ( d_1to @ X2 )
@ ^ [X4: $i] : ( ap @ X3 @ ( right1to @ X1 @ X2 @ X4 ) ) ) ) ) ).
thf(typ_left_f1,type,
left_f1: $i > $i > $i > $i > $i ).
thf(def_left_f1,definition,
( left_f1
= ( ^ [X0: $i,X1: $i,X2: $i] : ( left @ X0 @ X2 @ X1 ) ) ) ).
thf(typ_left_f2,type,
left_f2: $i > $i > $i > $i > $i ).
thf(def_left_f2,definition,
( left_f2
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] : ( left @ X0 @ X1 @ X2 @ ( left_f1 @ X0 @ X1 @ X2 @ X3 ) ) ) ) ).
thf(typ_frac,type,
frac: $i ).
thf(def_frac,definition,
( frac
= ( pair1type @ nat ) ) ).
thf(typ_n_fr,type,
n_fr: $i > $i > $i ).
thf(def_n_fr,definition,
( n_fr
= ( pair1 @ nat ) ) ).
thf(typ_num,type,
num: $i > $i ).
thf(def_num,definition,
( num
= ( first1 @ nat ) ) ).
thf(typ_den,type,
den: $i > $i ).
thf(def_den,definition,
( den
= ( second1 @ nat ) ) ).
thf(typ_n_eq,type,
n_eq: $i > $i > $o ).
thf(def_n_eq,definition,
( n_eq
= ( ^ [X0: $i,X1: $i] : ( n_is @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) ) ) ).
thf(satz37,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] : ( n_eq @ X0 @ X0 ) ) ).
thf(satz38,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( n_eq @ X0 @ X1 )
=> ( n_eq @ X1 @ X0 ) ) ) ) ).
thf(satz39,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( n_eq @ X0 @ X1 )
=> ( ( n_eq @ X1 @ X2 )
=> ( n_eq @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz40,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_eq @ X0 @ ( n_fr @ ( n_ts @ ( num @ X0 ) @ X1 ) @ ( n_ts @ ( den @ X0 ) @ X1 ) ) ) ) ) ).
thf(satz40a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ nat )
@ ^ [X1: $i] : ( n_eq @ ( n_fr @ ( n_ts @ ( num @ X0 ) @ X1 ) @ ( n_ts @ ( den @ X0 ) @ X1 ) ) @ X0 ) ) ) ).
thf(satz40b,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_eq @ ( n_fr @ X0 @ X1 ) @ ( n_fr @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz40c,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_eq @ ( n_fr @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) @ ( n_fr @ X0 @ X1 ) ) ) ) ) ).
thf(typ_moref,type,
moref: $i > $i > $o ).
thf(def_moref,definition,
( moref
= ( ^ [X0: $i,X1: $i] : ( d_29_ii @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) ) ) ).
thf(typ_lessf,type,
lessf: $i > $i > $o ).
thf(def_lessf,definition,
( lessf
= ( ^ [X0: $i,X1: $i] : ( iii @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) ) ) ).
thf(satz41,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( orec3 @ ( n_eq @ X0 @ X1 ) @ ( moref @ X0 @ X1 ) @ ( lessf @ X0 @ X1 ) ) ) ) ).
thf(satz41a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( or3 @ ( n_eq @ X0 @ X1 ) @ ( moref @ X0 @ X1 ) @ ( lessf @ X0 @ X1 ) ) ) ) ).
thf(satz41b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( ec3 @ ( n_eq @ X0 @ X1 ) @ ( moref @ X0 @ X1 ) @ ( lessf @ X0 @ X1 ) ) ) ) ).
thf(satz42,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( moref @ X0 @ X1 )
=> ( lessf @ X1 @ X0 ) ) ) ) ).
thf(satz43,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( lessf @ X0 @ X1 )
=> ( moref @ X1 @ X0 ) ) ) ) ).
thf(satz44,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( moref @ X0 @ X1 )
=> ( ( n_eq @ X0 @ X2 )
=> ( ( n_eq @ X1 @ X3 )
=> ( moref @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz45,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( lessf @ X0 @ X1 )
=> ( ( n_eq @ X0 @ X2 )
=> ( ( n_eq @ X1 @ X3 )
=> ( lessf @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(typ_moreq,type,
moreq: $i > $i > $o ).
thf(def_moreq,definition,
( moreq
= ( ^ [X0: $i,X1: $i] : ( l_or @ ( moref @ X0 @ X1 ) @ ( n_eq @ X0 @ X1 ) ) ) ) ).
thf(typ_lesseq,type,
lesseq: $i > $i > $o ).
thf(def_lesseq,definition,
( lesseq
= ( ^ [X0: $i,X1: $i] : ( l_or @ ( lessf @ X0 @ X1 ) @ ( n_eq @ X0 @ X1 ) ) ) ) ).
thf(satz41c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( moreq @ X0 @ X1 )
=> ( d_not @ ( lessf @ X0 @ X1 ) ) ) ) ) ).
thf(satz41d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( lesseq @ X0 @ X1 )
=> ( d_not @ ( moref @ X0 @ X1 ) ) ) ) ) ).
thf(satz41e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( d_not @ ( moref @ X0 @ X1 ) )
=> ( lesseq @ X0 @ X1 ) ) ) ) ).
thf(satz41f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( d_not @ ( lessf @ X0 @ X1 ) )
=> ( moreq @ X0 @ X1 ) ) ) ) ).
thf(satz41g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( moref @ X0 @ X1 )
=> ( d_not @ ( lesseq @ X0 @ X1 ) ) ) ) ) ).
thf(satz41h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( lessf @ X0 @ X1 )
=> ( d_not @ ( moreq @ X0 @ X1 ) ) ) ) ) ).
thf(satz41j,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( d_not @ ( moreq @ X0 @ X1 ) )
=> ( lessf @ X0 @ X1 ) ) ) ) ).
thf(satz41k,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( d_not @ ( lesseq @ X0 @ X1 ) )
=> ( moref @ X0 @ X1 ) ) ) ) ).
thf(satz46,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( moreq @ X0 @ X1 )
=> ( ( n_eq @ X0 @ X2 )
=> ( ( n_eq @ X1 @ X3 )
=> ( moreq @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz47,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( lesseq @ X0 @ X1 )
=> ( ( n_eq @ X0 @ X2 )
=> ( ( n_eq @ X1 @ X3 )
=> ( lesseq @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz48,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( moreq @ X0 @ X1 )
=> ( lesseq @ X1 @ X0 ) ) ) ) ).
thf(satz49,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( lesseq @ X0 @ X1 )
=> ( moreq @ X1 @ X0 ) ) ) ) ).
thf(satz50,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lessf @ X0 @ X1 )
=> ( ( lessf @ X1 @ X2 )
=> ( lessf @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz51a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lesseq @ X0 @ X1 )
=> ( ( lessf @ X1 @ X2 )
=> ( lessf @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz51b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lessf @ X0 @ X1 )
=> ( ( lesseq @ X1 @ X2 )
=> ( lessf @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz51c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( moreq @ X0 @ X1 )
=> ( ( moref @ X1 @ X2 )
=> ( moref @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz51d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( moref @ X0 @ X1 )
=> ( ( moreq @ X1 @ X2 )
=> ( moref @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz52,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lesseq @ X0 @ X1 )
=> ( ( lesseq @ X1 @ X2 )
=> ( lesseq @ X0 @ X2 ) ) ) ) ) ) ).
thf(satz53,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( l_some @ frac
@ ^ [X1: $i] : ( moref @ X1 @ X0 ) ) ) ).
thf(satz54,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( l_some @ frac
@ ^ [X1: $i] : ( lessf @ X1 @ X0 ) ) ) ).
thf(satz55,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( lessf @ X0 @ X1 )
=> ( l_some @ frac
@ ^ [X2: $i] : ( d_and @ ( lessf @ X0 @ X2 ) @ ( lessf @ X2 @ X1 ) ) ) ) ) ) ).
thf(typ_n_pf,type,
n_pf: $i > $i > $i ).
thf(def_n_pf,definition,
( n_pf
= ( ^ [X0: $i,X1: $i] : ( n_fr @ ( n_pl @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) @ ( n_ts @ ( den @ X0 ) @ ( den @ X1 ) ) ) ) ) ).
thf(satz56,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( n_eq @ X0 @ X1 )
=> ( ( n_eq @ X2 @ X3 )
=> ( n_eq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz57,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_eq @ ( n_pf @ ( n_fr @ X0 @ X2 ) @ ( n_fr @ X1 @ X2 ) ) @ ( n_fr @ ( n_pl @ X0 @ X1 ) @ X2 ) ) ) ) ) ).
thf(satz57a,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_eq @ ( n_fr @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pf @ ( n_fr @ X0 @ X2 ) @ ( n_fr @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz58,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( n_eq @ ( n_pf @ X0 @ X1 ) @ ( n_pf @ X1 @ X0 ) ) ) ) ).
thf(satz59,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] : ( n_eq @ ( n_pf @ ( n_pf @ X0 @ X1 ) @ X2 ) @ ( n_pf @ X0 @ ( n_pf @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz60,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( moref @ ( n_pf @ X0 @ X1 ) @ X0 ) ) ) ).
thf(satz60a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( lessf @ X0 @ ( n_pf @ X0 @ X1 ) ) ) ) ).
thf(satz61,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( moref @ X0 @ X1 )
=> ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz62b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( n_eq @ X0 @ X1 )
=> ( n_eq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz62c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lessf @ X0 @ X1 )
=> ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) ) ) ) ) ) ).
thf(satz62d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( moref @ X0 @ X1 )
=> ( moref @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz62e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( n_eq @ X0 @ X1 )
=> ( n_eq @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz62f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lessf @ X0 @ X1 )
=> ( lessf @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) ) ) ) ) ) ).
thf(satz62g,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( n_eq @ X0 @ X1 )
=> ( ( moref @ X2 @ X3 )
=> ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz62h,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( n_eq @ X0 @ X1 )
=> ( ( moref @ X2 @ X3 )
=> ( moref @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz62j,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( n_eq @ X0 @ X1 )
=> ( ( lessf @ X2 @ X3 )
=> ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz62k,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( n_eq @ X0 @ X1 )
=> ( ( lessf @ X2 @ X3 )
=> ( lessf @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X3 @ X1 ) ) ) ) ) ) ) ) ).
thf(satz63a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) )
=> ( moref @ X0 @ X1 ) ) ) ) ) ).
thf(satz63b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( n_eq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) )
=> ( n_eq @ X0 @ X1 ) ) ) ) ) ).
thf(satz63c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) )
=> ( lessf @ X0 @ X1 ) ) ) ) ) ).
thf(satz63d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( moref @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) )
=> ( moref @ X0 @ X1 ) ) ) ) ) ).
thf(satz63e,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( n_eq @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) )
=> ( n_eq @ X0 @ X1 ) ) ) ) ) ).
thf(satz63f,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( ( lessf @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) )
=> ( lessf @ X0 @ X1 ) ) ) ) ) ).
thf(satz64,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( moref @ X0 @ X1 )
=> ( ( moref @ X2 @ X3 )
=> ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz64a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( lessf @ X0 @ X1 )
=> ( ( lessf @ X2 @ X3 )
=> ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz65a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( moreq @ X0 @ X1 )
=> ( ( moref @ X2 @ X3 )
=> ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz65b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( moref @ X0 @ X1 )
=> ( ( moreq @ X2 @ X3 )
=> ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz65c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( lesseq @ X0 @ X1 )
=> ( ( lessf @ X2 @ X3 )
=> ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz65d,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( lessf @ X0 @ X1 )
=> ( ( lesseq @ X2 @ X3 )
=> ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz66,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( moreq @ X0 @ X1 )
=> ( ( moreq @ X2 @ X3 )
=> ( moreq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz66a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( lesseq @ X0 @ X1 )
=> ( ( lesseq @ X2 @ X3 )
=> ( lesseq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(satz67b,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X2: $i] : ( in @ X2 @ frac )
@ ^ [X2: $i] :
( all_of
@ ^ [X3: $i] : ( in @ X3 @ frac )
@ ^ [X3: $i] :
( ( n_eq @ ( n_pf @ X1 @ X2 ) @ X0 )
=> ( ( n_eq @ ( n_pf @ X1 @ X3 ) @ X0 )
=> ( n_eq @ X2 @ X3 ) ) ) ) ) ) ) ).
thf(typ_d_367_vo,type,
d_367_vo: $i > $i > $i ).
thf(def_d_367_vo,definition,
( d_367_vo
= ( ^ [X0: $i,X1: $i] : ( ind @ nat @ ( diffprop @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) ) ) ) ).
thf(typ_d_367_w,type,
d_367_w: $i > $i > $i ).
thf(def_d_367_w,definition,
( d_367_w
= ( ^ [X0: $i,X1: $i] : ( n_fr @ ( d_367_vo @ X0 @ X1 ) @ ( n_ts @ ( den @ X0 ) @ ( den @ X1 ) ) ) ) ) ).
thf(satz67a,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( moref @ X0 @ X1 )
=> ( l_some @ frac
@ ^ [X2: $i] : ( n_eq @ ( n_pf @ X1 @ X2 ) @ X0 ) ) ) ) ) ).
thf(k_satz67c,axiom,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( moref @ X0 @ X1 )
=> ( n_eq @ ( n_pf @ X1 @ ( d_367_w @ X0 @ X1 ) ) @ X0 ) ) ) ) ).
thf(satz67d,conjecture,
( all_of
@ ^ [X0: $i] : ( in @ X0 @ frac )
@ ^ [X0: $i] :
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( ( moref @ X0 @ X1 )
=> ( n_eq @ X0 @ ( n_pf @ X1 @ ( d_367_w @ X0 @ X1 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------