### Adjacency Concept for Pairs of Natural Numbers

by
Yatsuka Nakamura, and
Andrzej Trybulec

Copyright (c) 1996 Association of Mizar Users

MML identifier: GOBRD10
[ MML identifier index ]

```environ

vocabulary ARYTM_1, FINSEQ_2, FINSEQ_1, FUNCT_1, RELAT_1, FUNCOP_1, MATRIX_1,
GOBRD10, FINSEQ_4;
notation TARSKI, ZFMISC_1, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, NAT_1,
BINARITH, RELAT_1, FUNCT_1, FINSEQ_1, FINSEQ_2, FINSEQ_4, MATRIX_1,
FUNCOP_1;
constructors BINARITH, MATRIX_1, FINSEQ_4, XREAL_0, MEMBERED;
clusters SUBSET_1, RELSET_1, XREAL_0, NAT_1, MEMBERED, NUMBERS, ORDINAL2;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
definitions TARSKI;
theorems TARSKI, NAT_1, FINSEQ_1, REAL_1, AXIOMS, FINSEQ_4, SQUARE_1,
BINARITH, FUNCT_1, FINSEQ_2, FUNCOP_1, FINSEQ_3, ZFMISC_1, XCMPLX_0,
XCMPLX_1;
schemes NAT_1, FINSEQ_1, FINSEQ_2;

begin

reserve i,j,k,k1,k2,n,m,i1,i2,j1,j2 for Nat,
x for set;

definition let i1,i2;
:Def1: i2=i1+1 or i1=i2+1;
symmetry;
irreflexivity by NAT_1:38;
end;

then i2=i1+1 or i1=i2+1 by Def1;
end;

theorem Th2:for i1,i2 st i1,i2 are_adjacent1 & 1<=i1 & 1<=i2
proof let i1,i2;assume A1:i1,i2 are_adjacent1 & 1<=i1 & 1<=i2;
then i2=i1+1 or i1=i2+1 by Def1;
then i2-1=i1 or i1-1=i2 by XCMPLX_1:26;
then A2:i2-1=i1-1+1 or i1-1=i2-1+1 by XCMPLX_1:27;
0<=i1-1 & 0<=i2-1 by A1,SQUARE_1:12;
then i1-'1=i1-1 & i2-'1=i2-1 by BINARITH:def 3;
end;

definition let i1,j1,i2,j2;
end;

theorem
then i1,i2 are_adjacent1 & j1=j2 or i1=i2 & j1,j2 are_adjacent1 by Def2;
or i1+1=i2+1 & j1+1,j2+1 are_adjacent1 by Th1;
end;

theorem
for i1,i2,j1,j2 st i1,j1,i2,j2 are_adjacent2 &
1<=i1 & 1<=i2 & 1<=j1 & 1<=j2 holds i1-'1,j1-'1,i2-'1,j2-'1 are_adjacent2
proof let i1,i2,j1,j2;assume A1:i1,j1,i2,j2 are_adjacent2 &
1<=i1 & 1<=i2 & 1<=j1 & 1<=j2;
then i1,i2 are_adjacent1 & j1=j2 or i1=i2 & j1,j2 are_adjacent1 by Def2;
or i1-'1=i2-'1 & j1-'1,j2-'1 are_adjacent1 by A1,Th2;
end;

definition let n, i;
redefine func n |-> i -> FinSequence of NAT means
:Def3: len it = n & for j st 1 <= j & j <= n holds it.j = i;
coherence by FINSEQ_2:77;
compatibility
proof
let a be FinSequence of NAT;
hereby assume A1: a = n |-> i;
hence len a = n by FINSEQ_2:69;
thus for j st 1 <= j & j <= n holds a.j = i
proof
let j; assume 1 <= j & j <= n;
then j in Seg n by FINSEQ_1:3;
hence thesis by A1,FINSEQ_2:70;
end;
end;
assume A2: len a = n & for j st 1 <= j & j <= n holds a.j = i;
then A3:   dom a = Seg n by FINSEQ_1:def 3;
now let z be set; assume A4:z in dom a;
then reconsider z' = z as Nat;
1 <= z' & z' <= n by A3,A4,FINSEQ_1:3;
hence a.z = i by A2;
end;
then a = (dom a) --> i by FUNCOP_1:17;
hence a = n |-> i by A3,FINSEQ_2:def 2;
end;
end;

canceled;

theorem
Th6:for n,i,j st i<=n & j<=n
holds ex fs1 being FinSequence of NAT st fs1.1=i & fs1.(len fs1)=j &
len fs1=(i-'j)+(j-'i)+1 &
(for k,k1 st 1<=k & k<=len fs1 & k1=fs1.k holds k1<=n) &
for i1 st 1<=i1 & i1<len fs1 holds
fs1.(i1+1)= (fs1/.i1)+1 or fs1.i1= (fs1/.(i1+1)) +1
proof let n,i,j;assume A1: i<=n & j<=n;
now per cases by REAL_1:def 5;
case A2:i<j; then j-i>0 by SQUARE_1:11; then A3:j-i+1>0+1 by REAL_1:53;
then A4:j-i+1>0 by AXIOMS:22;
deffunc F(Nat) = i+\$1-'1;
ex p being FinSequence st len p = (j+1)-'i &
for k st k in Seg ((j+1)-'i) holds p.k=F(k) from SeqLambda;
then consider p being FinSequence such that
A5:len p = (j+1)-'i &
for k st k in Seg ((j+1)-'i) holds p.k=i+k-'1;
A6:for k,k1 st 1<=k & k<=len p & k1=p.k holds k1<=n
proof let k,k1;assume A7:1<=k & k<=len p & k1=p.k;
then k in Seg len p by FINSEQ_1:3;
then A8:k1=k+i-'1 by A5,A7;
A9:k+i<=(j+1)-'i +i by A5,A7,AXIOMS:24;
j+1-i>0 by A4,XCMPLX_1:29;
then k+i<=j+1-i+i by A9,BINARITH:def 3;
then A10:k+i<=j+1 by XCMPLX_1:27;
k+i-1=k+(i-1) by XCMPLX_1:29;
then 1+(i-1)<=k+i-1 by A7,AXIOMS:24;
then A11:i<=k+i-1 by XCMPLX_1:27; 0<=i by NAT_1:18;
then k1=k+i-1 by A8,A11,BINARITH:def 3;
then k1<=j+1-1 by A10,REAL_1:49;
then k1<=j by XCMPLX_1:26;
hence k1<=n by A1,AXIOMS:22;
end;
j+1-i>=0 by A4,XCMPLX_1:29;
then (j+1)-'i=j+1-i by BINARITH:def 3;
then A12:(j+1)-'i=j-i+1 by XCMPLX_1:29;
then 1 in Seg (j+1-'i) by A3,FINSEQ_1:3;
then p.1=i+1-'1 by A5;
then A13:p.1=i by BINARITH:39;
j-i>=i-i by A2,REAL_1:49;
then A14:j-i>=0 by XCMPLX_1:14;
then A15:j+1-'i=j-'i+1 by A12,BINARITH:def 3;
then A16:(j+1-'i) in Seg (j+1-'i) by FINSEQ_1:6;
A17:i+(j+1-'i)= (i+(j-'i))+1 by A15,XCMPLX_1:1;
1<=1+(i+(j-'i)) by NAT_1:29;
then i+(j+1-'i)-1>=1-1 by A17,REAL_1:49;
then i+(j+1-'i)-'1=i+(j+1-'i)-1 by BINARITH:def 3;
then A18: p.(len p)=(i+(j-i+1))-1 by A5,A12,A16
.=(i+(j-i)+1)-1 by XCMPLX_1:1 .=(j+1)-1 by XCMPLX_1:27
.=j by XCMPLX_1:26;
i-j<j-j by A2,REAL_1:54;
then i-j<0 by XCMPLX_1:14; then i-'j=0 by BINARITH:def 3;
then A19:len p=(i-'j)+(j-'i)+1 by A5,A12,A14,BINARITH:def 3;
rng p c=NAT
proof let x;assume x in rng p;
then consider y being set such that
A20: y in dom p & p.y=x by FUNCT_1:def 5;
A21:y in Seg len p by A20,FINSEQ_1:def 3;
then y in {k:1<=k & k<=len p} by FINSEQ_1:def 1;
then consider k such that A22: k=y & (1<=k & k<=len p);
p.k=i+k-'1 by A5,A21,A22;
hence thesis by A20,A22;
end;
then reconsider fs2=p as FinSequence of NAT by FINSEQ_1:def 4;
for i1 st 1<=i1 & i1<len fs2 holds
fs2.(i1+1)= (fs2/.i1)+1 or fs2.i1= (fs2/.(i1+1)) +1
proof let i1;assume A23:1<=i1 & i1<len fs2;
then A24:i1+1<=len fs2 by NAT_1:38; 1<=1+i1 by NAT_1:29;
then i1+1 in Seg len fs2 by A24,FINSEQ_1:3;
then A25:fs2.(i1+1)=i+(i1+1)-'1 by A5;
A26:    i1 in Seg len fs2 by A23,FINSEQ_1:3;
A27: fs2/.i1 = fs2.i1 by A23,FINSEQ_4:24;
0<=i by NAT_1:18;
then i+i1>=1+0 by A23,REAL_1:55;
then i+i1-1>=1-1 by REAL_1:49;
then A28: 1+(i+i1-'1)=1+(i+i1-1) by BINARITH:def 3
.=i+i1 by XCMPLX_1:27;
A29:fs2.(i1+1)=1+(i+i1)-'1 by A25,XCMPLX_1:1;
1+(i+i1)>=1 by NAT_1:29;
then 1+(i+i1)-1>=1-1 by REAL_1:49;
then (1+(i+i1))-'1=(1+(i+i1))-1 by BINARITH:def 3
.=i+i1 by XCMPLX_1:26;
hence thesis by A5,A26,A27,A28,A29;
end;
hence thesis by A6,A13,A18,A19;
case A30:i=j;
consider f being Function such that
A31: dom f=Seg 1 & rng f={i} by FINSEQ_1:4,FUNCT_1:15;
A32:rng f c=NAT by A31,ZFMISC_1:37;
f is FinSequence-like by A31,FINSEQ_1:def 2;
then reconsider fs1=f as FinSequence of NAT by A32,FINSEQ_1:def 4;
1 in dom f by A31,FINSEQ_1:3;
then fs1.1 in rng f by FUNCT_1:def 5;
then A33:fs1.1=i by A31,TARSKI:def 1;
A34:len fs1=1 by A31,FINSEQ_1:def 3;
i-j=0 by A30,XCMPLX_1:14;
then A35:i-'j=0 by BINARITH:def 3;
A36:(for k,k1 st 1<=k & k<=len fs1 & k1=fs1.k holds k1<=n)
by A1,A33,A34,AXIOMS:21;
for i1 st 1<=i1 & i1<len fs1 holds
fs1.(i1+1)=(fs1/.i1)+1 or fs1.i1=(fs1/.(i1+1)) +1 by A31,FINSEQ_1:def 3;
hence thesis by A30,A33,A34,A35,A36;
case A37:j<i; then i-j>0 by SQUARE_1:11;
then A38: i-j+1>0+1 by REAL_1:53;
then A39:i+1-j>=1 by XCMPLX_1:29;
i-j+1>0 by A38,AXIOMS:22;
then A40:i+1-j>=0 by XCMPLX_1:29;
deffunc F(Nat) = i+1-'\$1;
ex p being FinSequence st len p = i+1-'j &
for k st k in Seg (i+1-'j) holds p.k=F(k) from SeqLambda;
then consider p being FinSequence such that
A41:len p = i+1-'j &
for k st k in Seg (i+1-'j) holds p.k=i+1-'k;
A42:for k,k1 st 1<=k & k<=len p & k1=p.k holds k1<=n
proof let k,k1;assume A43:1<=k & k<=len p & k1=p.k;
then k in Seg ((i+1)-'j) by A41,FINSEQ_1:3;
then A44:k1=i+1-'k by A41,A43;
-k<=-1 by A43,REAL_1:50;
then -k+(i+1)<= -1+(i+1) by AXIOMS:24;
then i+1-k<=i+1+-1 by XCMPLX_0:def 8;
then A45:i+1-k<=i+1-1 by XCMPLX_0:def 8;
k<=i+1-j by A40,A41,A43,BINARITH:def 3;
then k+j<=i+1-j+j by AXIOMS:24; then k+j<=i+1 by XCMPLX_1:27;
then k+j-k<=i+1-k by REAL_1:49;
then A46:j<=i+1-k by XCMPLX_1:26; 0<=j by NAT_1:18;
then k1=i+1-k by A44,A46,BINARITH:def 3;
then k1<=i by A45,XCMPLX_1:26;
hence k1<=n by A1,AXIOMS:22;
end;
A47:i+1-'j=i+1-j by A40,BINARITH:def 3;
then 1 in Seg (i+1-'j) by A39,FINSEQ_1:3;
then A48:p.1=i+1-'1 by A41;
i+1-1=i by XCMPLX_1:26;
then i+1-1>=0 by NAT_1:18;
then A49:p.1=i+1-1 by A48,BINARITH:def 3.=i by XCMPLX_1:26;
A50: i-j>=j-j by A37,REAL_1:49;
then A51:i-j>=0 by XCMPLX_1:14;
then i-j+1>=0+1 by AXIOMS:24;
then A52:i+1-j>=1 by XCMPLX_1:29;
then A53:i+1-j>=0 by AXIOMS:22;
then A54:i+1-'j=i+1-j by BINARITH:def 3;
then A55:(i+1-'j) in Seg (i+1-'j) by A52,FINSEQ_1:3;
i+1-(i+1-'j)=i+1-(i+1-j) by A40,BINARITH:def 3
.=j by XCMPLX_1:18;
then A56:i+1-(i+1-'j)>=0 by NAT_1:18;
A57: now per cases by A50,XCMPLX_1:14;
case i-j=0; then -(i-j)=-0;
then j-i=0 by XCMPLX_1:143;
hence j-'i=0 by BINARITH:def 3;
case i-j>0; then --(i-j)>0; then -(i-j)<0 by REAL_1:66;
then j-i<0 by XCMPLX_1:143;
hence j-'i=0 by BINARITH:def 3;
end;
A58:p.(len p)=(i+1-'(i+1-'j)) by A41,A55
.= (i+1-(i+1-'j)) by A56,BINARITH:def 3
.= i+1-(i+1-j) by A53,BINARITH:def 3
.= j by XCMPLX_1:18;
j-i<i-i by A37,REAL_1:54;
then j-i<0 by XCMPLX_1:14; then j-'i=0 by BINARITH:def 3;
then A59:(i-'j)+(j-'i)+1
=i-j+1+(j-'i) by A51,BINARITH:def 3
.=i+1-j by A57,XCMPLX_1:29;
rng p c=NAT
proof let x;assume x in rng p;
then consider y being set such that
A60: y in dom p & p.y=x by FUNCT_1:def 5;
A61:y in Seg len p by A60,FINSEQ_1:def 3;
then y in {k:1<=k & k<=len p} by FINSEQ_1:def 1;
then consider k such that A62: k=y & 1<=k & k<=len p;
p.k=i+1-'k by A41,A61,A62;
hence thesis by A60,A62;
end;
then reconsider fs2=p as FinSequence of NAT by FINSEQ_1:def 4;
for i1 st 1<=i1 & i1<len fs2 holds
fs2.(i1+1)= (fs2/.i1)+1 or fs2.i1= (fs2/.(i1+1)) +1
proof let i1;assume A63:1<=i1 & i1<len fs2;
then i1 in Seg len fs2 by FINSEQ_1:3;
then A64:fs2.(i1)=i+1-'i1 by A41;
A65:1<=i1+1 & i1+1<=len fs2 by A63,NAT_1:38;
then (i1+1) in Seg len fs2 by FINSEQ_1:3;
then fs2.(i1+1)=i+1-'(i1+1) by A41;
then A66:(fs2/.(i1+1))+1=1+(i+1-'(i1+1)) by A65,FINSEQ_4:24;
i1+1<=i-j+1 by A41,A54,A65,XCMPLX_1:29;
then i1<=i-j by REAL_1:53;
then i1+j<=i by REAL_1:84;
then A67:j<=i-i1 by REAL_1:84;
0<=j by NAT_1:18; then 0+1<=i-i1+1 by A67,AXIOMS:24;
then A68:1<=i+1-i1 by XCMPLX_1:29; then 1-1<=i+1-i1-1 by REAL_1:49;
then i+1-(i1+1)>=0 by XCMPLX_1:36;
then A69: 1+(i+1-'(i1+1))=1+(i+1-(i1+1)) by BINARITH:def 3
.=1+(i+1-i1-1) by XCMPLX_1:36 .=i+1-i1 by XCMPLX_1:27;
0<=i+1-i1 by A68,AXIOMS:22;
hence thesis by A64,A66,A69,BINARITH:def 3;
end;
hence thesis by A41,A42,A47,A49,A58,A59;
end;
hence thesis;
end;

theorem
Th7:for n,i,j st i<=n & j<=n
holds ex fs1 being FinSequence of NAT st fs1.1=i & fs1.(len fs1)=j &
len fs1=(i-'j)+(j-'i)+1 &
(for k,k1 st 1<=k & k<=len fs1 & k1=fs1.k holds k1<=n) &
for i1 st 1<=i1 & i1<len fs1 holds fs1/.i1,fs1/.(i1+1) are_adjacent1
proof let n,i,j;assume i<=n & j<=n;
then consider fs1 being FinSequence of NAT such that
A1: fs1.1=i & fs1.(len fs1)=j &
len fs1=(i-'j)+(j-'i)+1 &
(for k,k1 st 1<=k & k<=len fs1 & k1=fs1.k holds k1<=n) &
for i1 st 1<=i1 & i1<len fs1 holds
fs1.(i1+1)= (fs1/.i1)+1 or fs1.i1= (fs1/.(i1+1)) +1 by Th6;
for i1 st 1<=i1 & i1<len fs1 holds
proof let i1;assume A2:1<=i1 & i1<len fs1;
then 1<= i1+1 & i1+1<=len fs1 by NAT_1:38;
then A3:fs1.(i1+1)= fs1/.(i1+1) by FINSEQ_4:24;
A4:fs1.i1= fs1/.i1 by A2,FINSEQ_4:24;
fs1.(i1+1)= (fs1/.i1)+1 or fs1.i1= (fs1/.(i1+1)) +1 by A1,A2;
hence thesis by A3,A4,Def1;
end;
hence thesis by A1;
end;

theorem
Th8:for n,m,i1,j1,i2,j2 st i1<=n & j1<=m & i2<=n & j2<=m
ex fs1,fs2 being FinSequence of NAT st
(for i,k1,k2 st i in dom fs1 & k1=fs1.i & k2=fs2.i
holds k1 <= n & k2 <= m) &
fs1.1=i1 & fs1.(len fs1)=i2 &
fs2.1=j1 & fs2.(len fs2)=j2 & len fs1=len fs2 &
len fs1=(i1-'i2)+(i2-'i1)+(j1-'j2)+(j2-'j1)+1 &
for i st 1<=i & i<len fs1 holds
proof let n,m,i1,j1,i2,j2;assume
A1: i1<=n & j1<=m & i2<=n & j2<=m;
then consider gs1 being FinSequence of NAT such that
A2:gs1.1=i1 & gs1.(len gs1)=i2 &
len gs1=(i1-'i2)+(i2-'i1)+1 &
(for k,k1 st 1<=k & k<=len gs1 & k1=gs1.k holds k1<=n) &
for k st 1<=k & k<len gs1 holds
consider gs2 being FinSequence of NAT such that
A3:gs2.1=j1 & gs2.(len gs2)=j2 &
len gs2=(j1-'j2)+(j2-'j1)+1 &
(for k,k1 st 1<=k & k<=len gs2 & k1=gs2.k holds k1<=m) &
for k st 1<=k & k<len gs2 holds
set hs1=gs1^((len gs2-'1) |-> i2), hs2=((len gs1-'1) |-> j1)^gs2;
A4:1<=1 & 1<=len gs2 by A3,NAT_1:29;
then A5:0<=len gs2-1 by SQUARE_1:12;
A6:1<=1 & 1<=len gs1 by A2,NAT_1:29;
then A7: 1 in dom gs1 by FINSEQ_3:27;
A8:0<=len gs1-1 by A6,SQUARE_1:12;
A9:   len gs2 -1>=1-1 by A4,REAL_1:49;
A10:   len gs1 -1>=1-1 by A6,REAL_1:49;
A11:len hs1=len gs1 + len ((len gs2-'1) |-> i2) by FINSEQ_1:35
.=len gs1 + (len gs2-'1) by Def3
.=len gs1 + (len gs2-1) by A9,BINARITH:def 3
.=len gs1 +len gs2 -1 by XCMPLX_1:29
.=(len gs1-1)+ len gs2 by XCMPLX_1:29;
A12: (len gs1-1) + len gs2
=(len gs1-'1)+ len gs2 by A10,BINARITH:def 3
.=len ((len gs1-'1) |-> j1) + len gs2 by Def3
.=len hs2 by FINSEQ_1:35;
A13: now per cases;
case len gs1-'1=0;
then ((len gs1-'1) |-> j1)^gs2=<*>(NAT)^gs2 by FINSEQ_2:72
.=gs2 by FINSEQ_1:47;
hence hs2.1=j1 & hs2.len hs2=j2 by A3;
case A14:len gs1-'1<>0;
A15:len ((len gs1-'1) |-> j1)=len gs1-'1 by Def3;
len gs1-'1>0 by A14,NAT_1:19;
then A16:      0+1<= len ((len gs1-'1) |-> j1) by A15,NAT_1:38;
then 1 in dom ((len gs1-'1) |-> j1) by FINSEQ_3:27;
then A17:hs2.1 = ((len gs1-'1) |-> j1).1 by FINSEQ_1:def 7;
A18: len ((len gs1-'1) |-> j1) = len gs1-'1 by Def3;
A19: len hs2=len ((len gs1-'1) |-> j1) + len gs2 by FINSEQ_1:35;
len gs2<=len gs2 & 1<=len gs2 by A3,NAT_1:29;
then len gs2 in dom gs2 by FINSEQ_3:27;
hence hs2.1=j1 & hs2.len hs2=j2 by A3,A16,A17,A18,A19,Def3,FINSEQ_1:
def 7;
end;
A20: now per cases;
case len gs2-'1=0;
then gs1^((len gs2-'1) |-> i2) = gs1^<*>(NAT) by FINSEQ_2:72
.=gs1 by FINSEQ_1:47;
hence hs1.1=i1 & hs1.len hs1=i2 by A2;
case len gs2-'1<>0;
then A21:len gs2-'1>0 by NAT_1:19;
A22: len hs1=len gs1+len ((len gs2-'1) |-> i2) by FINSEQ_1:35;
A23:len ((len gs2-'1) |-> i2) = len gs2-'1 by Def3;
A24:      0+1<=len gs2-'1 by A21,NAT_1:38;
then A25:len ((len gs2-'1) |-> i2) in dom ((len gs2-'1) |-> i2)
by A23,FINSEQ_3:27;
((len gs2-'1) |-> i2).len ((len gs2-'1) |-> i2) = i2
by A23,A24,Def3;
hence hs1.1=i1 & hs1.len hs1=i2 by A2,A7,A22,A25,FINSEQ_1:def 7;
end;
A26:(i1-'i2)+(i2-'i1)+(j1-'j2)+(j2-'j1)+1
= len gs1 -1+ (j1-'j2)+(j2-'j1)+1 by A2,XCMPLX_1:26
.= len gs1 -1+ ((j1-'j2)+(j2-'j1))+1 by XCMPLX_1:1
.= len hs1 by A3,A11,XCMPLX_1:1;
A27:for i,k1,k2 st i in dom hs1 & k1=hs1.i & k2=hs2.i
holds k1<=n & k2<=m
proof let i,k1,k2;assume A28:i in dom hs1 & k1=hs1.i & k2=hs2.i;
then A29:i in Seg len hs1 by FINSEQ_1:def 3;
dom hs1=Seg(len gs1 + len ((len gs2-'1) |-> i2))
by FINSEQ_1:def 7;
then A30: len hs1=len gs1 + len ((len gs2-'1) |-> i2)
by FINSEQ_1:def 3;
then A31:len hs1=len gs1 + (len gs2 -'1) by Def3;
len gs1<=len hs1 by A30,NAT_1:29;
then len hs1 -len gs1>=0 by SQUARE_1:12;
then A32:len hs1 -' len gs1 = len hs1 - len gs1 by BINARITH:def 3;
A33: 1<=i & i<=len gs1 or
len gs1 -len gs1<i-len gs1 & i <=len hs1 by A28,FINSEQ_3:27,REAL_1:
54;
A34: now per cases by A32,A33,REAL_1:49,XCMPLX_1:14;
case A35:1<=i & i<=len gs1;
then i in dom gs1 by FINSEQ_3:27;
then hs1.i=gs1.i by FINSEQ_1:def 7;
hence k1<=n by A2,A28,A35;
case A36:0<i-len gs1 & i-len gs1 <=len hs1 -' len gs1;
then A37:i-len gs1=i-'len gs1 by BINARITH:def 3;
then 0+1<=i-'len gs1 by A36,NAT_1:38;
then (i-'len gs1) in Seg (len hs1-'len gs1)
by A36,A37,FINSEQ_1:3;
then A38:(i-'len gs1)in Seg (len gs2-'1) by A31,BINARITH:39;
then (i-'len gs1)in Seg len ((len gs2-'1) |-> i2) by Def3;
then A39:(i-'len gs1) in dom ((len gs2-'1) |-> i2) by FINSEQ_1:def 3;
A40:1<=i-'len gs1 & i-'len gs1<=len gs2 -'1 by A38,FINSEQ_1:3;
A41:i-'len gs1=i- len gs1 by A36,BINARITH:def 3;
i=len gs1+(i-len gs1) by XCMPLX_1:27;
then hs1.i= ((len gs2-'1) |-> i2).(i-'len gs1) by A39,A41,FINSEQ_1:
def 7
.=i2 by A40,Def3;
hence k1<=n by A1,A28;
end;
dom hs2=Seg(len ((len gs1-'1) |-> j1)+ len gs2)
by FINSEQ_1:def 7;
then len hs2=len ((len gs1-'1) |-> j1) + len gs2 by FINSEQ_1:def 3;
then A42:len hs2=(len gs1 -'1)+len gs2 by Def3;
1<=i & i<=len gs1-'1 or (len gs1 -'1)+1<=i & i<=len hs2
by A11,A12,A29,FINSEQ_1:3,NAT_1:38;
then 1<=i & i<=len gs1-'1 or len gs1-1+1<=i & i<=len hs2
by A10,BINARITH:def 3;
then 1<=i & i<=len gs1-1 or len gs1<=i & i<=len hs2
by A10,BINARITH:def 3,XCMPLX_1:27;
then 1<=i & i<=len gs1-1 or
len gs1 -(len gs1-1)<=i-(len gs1-1) & i <=len hs2 by REAL_1:49;
then 1<=i & i<=len gs1-1 or
1<=i-(len gs1-1) & i-(len gs1-1) <=len hs2-(len gs1-1)
by REAL_1:49,XCMPLX_1:18;
then 1<=i & i<=len gs1-1 or
1<=i-len gs1+1 & i-len gs1+1 <=len hs2-(len gs1-1)
by XCMPLX_1:37;
then A43: 1<=i & i<=len gs1-1 or
1<=i-len gs1+1 & i-len gs1+1 <=len hs2-len gs1+1
by XCMPLX_1:37;
now per cases by A43,REAL_1:53;
case A44:1<=i-len gs1+1 & i-len gs1 <=len hs2 - len gs1;
then A45:1<=i+1-len gs1 by XCMPLX_1:29;
then A46:0<=i+1-len gs1 by AXIOMS:22;
i-len gs1+1<=len hs2 - len gs1+1 by A44,AXIOMS:24;
then i+1-len gs1<=len hs2 - len gs1+1 by XCMPLX_1:29;
then A47:i+1-len gs1<=len hs2+1 - len gs1 by XCMPLX_1:29;
A48: len hs2+1 - len gs1=(len gs1 -1)+len gs2+1 - len gs1
by A10,A42,BINARITH:def 3
.=(len gs1 -1)+len gs2+(1-len gs1) by XCMPLX_1:29
.=(len gs1 -1)+len gs2-(len gs1-1) by XCMPLX_1:38
.=len gs2 by XCMPLX_1:26;
A49:i+1-len gs1=i+1-'len gs1 by A46,BINARITH:def 3;
A50:len ((len gs1-'1) |-> j1)=len gs1-'1 by Def3;
i+1-'len gs1 in Seg len gs2 by A45,A47,A48,A49,FINSEQ_1:3;
then A51:i+1-'len gs1 in dom gs2 by FINSEQ_1:def 3;
i=len gs1-1+(i-(len gs1-1)) by XCMPLX_1:27
.=len gs1-1+(i-len gs1+1) by XCMPLX_1:37
.=len gs1-1+(i+1-len gs1) by XCMPLX_1:29
.=(len ((len gs1-'1) |-> j1)+(i+1-'len gs1))
by A10,A49,A50,BINARITH:def 3;
then hs2.i= gs2.(i+1-'len gs1) by A51,FINSEQ_1:def 7;
hence k2<=m by A3,A28,A45,A47,A48,A49;
case 1<=i & i<=len gs1-1;
then A52:1<=i & i<=len gs1-'1 by A10,BINARITH:def 3;
then i in Seg (len gs1 -'1) by FINSEQ_1:3;
then i in Seg len ((len gs1-'1) |-> j1) by Def3;
then i in dom ((len gs1-'1) |-> j1) by FINSEQ_1:def 3;
then hs2.i= ((len gs1-'1) |-> j1).i by FINSEQ_1:def 7
.=j1 by A52,Def3;
hence k2<=m by A1,A28;
end;
hence k1<=n & k2<=m by A34;
end;
for i st 1<=i & i<len hs1 holds
proof let i;assume A53:1<=i & i<len hs1;
now per cases;
case A54: i<len gs1; then A55:i+1<=len gs1 by NAT_1:38;
A56:1<=i+1 by A53,NAT_1:38;
i+1-1<=len gs1 -1 by A55,REAL_1:49;
then i<=len gs1-1 by XCMPLX_1:26;
then A57:1<=i & i<=len gs1-'1 by A10,A53,BINARITH:def 3;
then A58:i in Seg (len gs1 -'1) by FINSEQ_1:3;
len ((len gs1-'1) |-> j1)=len gs1-'1 by Def3;
then A59:i in dom ((len gs1-'1) |-> j1) by A58,FINSEQ_1:def 3;
A60: dom ((len gs1-'1) |-> j1) c= dom hs2 by FINSEQ_1:39;
A61: hs2.i= ((len gs1-'1) |-> j1).i by A59,FINSEQ_1:def 7
.=j1 by A57,Def3;
A62: now per cases;
case A63:i+1<=len gs1-'1;
then A64:i+1 in Seg (len gs1 -'1) by A56,FINSEQ_1:3;
len ((len gs1-'1) |-> j1)=len gs1-'1 by Def3;
then A65:i+1 in dom ((len gs1-'1) |-> j1) by A64,FINSEQ_1:def 3;
then A66:hs2.(i+1)= ((len gs1-'1) |-> j1).(i+1) by FINSEQ_1:def 7
.=j1 by A56,A63,Def3;
dom ((len gs1-'1) |-> j1) c= dom hs2 by FINSEQ_1:39;
hence hs2/.(i+1)=j1 by A65,A66,FINSEQ_4:def 4;
case i+1>len gs1-'1;
then A67:len gs1-'1+1 <=i+1 by NAT_1:38;
A68:len gs1-'1+1=len gs1-1+1 by A8,BINARITH:def 3
.=len gs1 by XCMPLX_1:27;
then A69:len gs1=i+1 by A55,A67,AXIOMS:21;
1 in Seg len gs2 by A4,FINSEQ_1:3;
then A70:1 in dom gs2 by FINSEQ_1:def 3;
len ((len gs1-'1) |-> j1)+1=i+1 by A68,A69,Def3;
then A71:hs2.(i+1)=j1 by A3,A70,FINSEQ_1:def 7;
dom hs2=Seg(len ((len gs1-'1) |-> j1)+len gs2)
by FINSEQ_1:def 7;
then len hs2=len ((len gs1-'1) |-> j1)+len gs2 by FINSEQ_1:def 3
.=(len gs1-'1)+len gs2 by Def3
.=len gs1-1+len gs2 by A8,BINARITH:def 3
.=len gs1+len gs2-1 by XCMPLX_1:29
.=len gs1+(len gs2-1) by XCMPLX_1:29
.=len gs1+(len gs2-'1) by A5,BINARITH:def 3;
then len gs1 <= len hs2 by NAT_1:29;
then len gs1 in Seg len hs2 by A6,FINSEQ_1:3;
then (i+1) in dom hs2 by A69,FINSEQ_1:def 3;
hence hs2/.(i+1)=j1 by A71,FINSEQ_4:def 4;
end;
now per cases;
case i+1<=len gs1-'1;
then i+1<=len gs1 -1 by A8,BINARITH:def 3;
then i+1+1<=len gs1-1+1 by AXIOMS:24;
then i+1+1<=len gs1 by XCMPLX_1:27;
then A72:      i+1<len gs1 by NAT_1:38;
A73:      1<i+1 by A53,NAT_1:38;
A74:i<len gs1 by A72,NAT_1:38;
A75: dom gs1 c= dom hs1 by FINSEQ_1:39;
A76:i in dom gs1 by A53,A74,FINSEQ_3:27;
A77: (i+1) in dom gs1 by A72,A73,FINSEQ_3:27;
A78:hs1/.i=hs1.i by A75,A76,FINSEQ_4:def 4
.=gs1.i by A76,FINSEQ_1:def 7
.=gs1/.i by A76,FINSEQ_4:def 4;
hs1/.(i+1)=hs1.(i+1) by A75,A77,FINSEQ_4:def 4
.=gs1.(i+1) by A77,FINSEQ_1:def 7
.=gs1/.(i+1) by A77,FINSEQ_4:def 4;
case i+1>len gs1-'1;
0<i+1 by NAT_1:19;
then A79: 0+1<=i+1 by NAT_1:38;
A80: dom gs1 c= dom hs1 by FINSEQ_1:39;
A81:i in dom gs1 by A53,A54,FINSEQ_3:27;
A82: (i+1) in dom gs1 by A55,A79,FINSEQ_3:27;
A83:hs1/.i=hs1.i by A80,A81,FINSEQ_4:def 4
.=gs1.i by A81,FINSEQ_1:def 7
.=gs1/.i by A81,FINSEQ_4:def 4;
hs1/.(i+1)=hs1.(i+1) by A80,A82,FINSEQ_4:def 4
.=gs1.(i+1) by A82,FINSEQ_1:def 7
.=gs1/.(i+1)by A82,FINSEQ_4:def 4;
end;
by A59,A60,A61,A62,FINSEQ_4:def 4;
case A84:i>=len gs1;
then A85:i-len gs1>=0 by SQUARE_1:12;
then A86:i-len gs1=i-'len gs1 by BINARITH:def 3;
i-len gs1+1>=0+1 by A85,AXIOMS:24;
then A87:i+1-len gs1>=1 by XCMPLX_1:29;
then A88: (i+1)-len gs1>=0 by AXIOMS:22;
A89:i in dom hs1 by A53,FINSEQ_3:27;
len hs1=len gs1+len ((len gs2-'1) |-> i2) by FINSEQ_1:35
.=len gs1 + (len gs2-'1) by Def3;
then i-len gs1 < len gs1 + (len gs2 -'1) - len gs1 by A53,REAL_1:54;
then i-len gs1 < len gs2 -'1 by XCMPLX_1:26;
then A90:(i-'len gs1)<=len gs2 -'1 by A85,BINARITH:def 3;
A91:now assume not 1<=(i-'len gs1);
then i-'len gs1<0+1;
then i-'len gs1<=0 by NAT_1:38;
then i-'len gs1=0 by NAT_1:18;
then A92:i=len gs1 by A86,XCMPLX_1:15;
0<len gs1 by A6,AXIOMS:22;
then len gs1 in Seg len gs1 by FINSEQ_1:5;
then i in dom gs1 by A92,FINSEQ_1:def 3;
hence hs1.i=i2 by A2,A92,FINSEQ_1:def 7;
end;
A93:len ((len gs2-'1) |-> i2)=len gs2-'1 by Def3;
now assume A94: 1<=(i-'len gs1);
then (i-'len gs1) in Seg len ((len gs2-'1) |-> i2)
by A90,A93,FINSEQ_1:3;
then A95:(i-'len gs1) in dom ((len gs2-'1) |-> i2) by FINSEQ_1:def 3;
i =len gs1+(i-len gs1) by XCMPLX_1:27
.=len gs1+(i-'len gs1) by A85,BINARITH:def 3;
then hs1.i= ((len gs2-'1) |-> i2).(i-'len gs1)by A95,FINSEQ_1:def 7
.=i2 by A90,A94,Def3;
hence hs1.i=i2;
end;
then A96:hs1/.i=i2 by A89,A91,FINSEQ_4:def 4;
A97:i+1<=len hs1 by A53,NAT_1:38;
0<i+1 by NAT_1:19;
then 0+1<=i+1 by NAT_1:38;
then A98:i+1 in Seg len hs1 by A97,FINSEQ_1:3;
then A99:i+1 in dom hs1 by FINSEQ_1:def 3;
len gs1<=i+1 by A84,NAT_1:37;
then A100:0<=i+1-len gs1 by SQUARE_1:12;
A101:len gs2-'1<=len gs2-'1+1 by NAT_1:29;
i+1<=len gs2+len gs1-1 by A11,A97,XCMPLX_1:29;
then i+1<=len gs2 -1+len gs1 by XCMPLX_1:29;
then i+1-len gs1+len gs1<=len gs2 -1+len gs1 by XCMPLX_1:27;
then i+1-len gs1<=len gs2 -1 by REAL_1:53;
then i+1-len gs1<=len gs2 -'1 by A5,BINARITH:def 3;
then A102:1<=(i+1-'len gs1) & (i+1-'len gs1)<=len gs2-'1
by A87,A88,BINARITH:def 3;
then 1<=(i+1-'len gs1) & (i+1-'len gs1)<=(len gs2-'1)+1
by A101,AXIOMS:22;
then 1<=(i+1-'len gs1) & (i+1-'len gs1)<=(len gs2-1)+1
by A9,BINARITH:def 3;
then A103:1<=(i+1-'len gs1) & (i+1-'len gs1)<=len gs2 by XCMPLX_1:27;
len ((len gs2-'1) |-> i2)=len gs2-'1 by Def3;
then (i+1-'len gs1) in Seg len ((len gs2-'1) |-> i2)
by A102,FINSEQ_1:3;
then A104:(i+1-'len gs1) in dom ((len gs2-'1) |-> i2) by FINSEQ_1:def
3;
i+1 =len gs1+(i+1-len gs1) by XCMPLX_1:27
.=len gs1+(i+1-'len gs1) by A100,BINARITH:def 3;
then A105: hs1.(i+1)=
((len gs2-'1) |-> i2).((i+1)-'len gs1) by A104,FINSEQ_1:def 7
.=i2 by A102,Def3;
0+1<=i-len gs1 +1 by A85,AXIOMS:24;
then 1<=i+1-len gs1 by XCMPLX_1:29;
then A106:1<=i+1-'len gs1 by A100,BINARITH:def 3;
A107:1<=i+1-'len gs1 +1 by NAT_1:29;
A108:i+1-'len gs1 in dom gs2 by A103,FINSEQ_3:27;
A109:i in dom hs2 by A11,A12,A53,FINSEQ_3:27;
A110:i+1 in dom hs2 by A11,A12,A98,FINSEQ_1:def 3;
i-(len gs1-1)<len gs1-1+len gs2 -(len gs1-1)
by A11,A53,REAL_1:54;
then i-(len gs1-1)<len gs2 by XCMPLX_1:26;
then i-len gs1+1<len gs2 by XCMPLX_1:37;
then i+1-len gs1<len gs2 by XCMPLX_1:29;
then A111:i+1-'len gs1<len gs2 by A100,BINARITH:def 3;
then len gs2>=i+1-'len gs1+1 by NAT_1:38;
then i+1-'len gs1+1 in Seg len gs2 by A107,FINSEQ_1:3;
then A112:i+1-'len gs1+1 in dom gs2 by FINSEQ_1:def 3;
len ((len gs1-'1) |-> j1)+(i+1-'len gs1+1)
=(len gs1-'1)+(i+1-'len gs1+1) by Def3
.=len gs1 -1+(i+1-'len gs1+1) by A10,BINARITH:def 3
.=len gs1 -1+(i+1-len gs1+1) by A100,BINARITH:def 3
.=len gs1 -1+(i+1-(len gs1-1)) by XCMPLX_1:37
.=i+1 by XCMPLX_1:27;
then A113:
hs2/.(i+1)=hs2.(len ((len gs1-'1) |-> j1)+(i+1-'len gs1+1)) by A110,FINSEQ_4:
def 4
.=gs2.((i+1-'len gs1+1))
by A112,FINSEQ_1:def 7
.=gs2/.(i+1-'len gs1+1) by A112,FINSEQ_4:def 4;
len ((len gs1-'1) |-> j1)+(i+1-'len gs1)
=(len gs1-'1)+(i+1-'len gs1) by Def3
.=len gs1 -1+(i+1-'len gs1) by A10,BINARITH:def 3
.=len gs1 -1+(i+1-len gs1) by A100,BINARITH:def 3
.=len gs1 -1+(i-len gs1+1) by XCMPLX_1:29
.=len gs1 -1+(i-(len gs1-1)) by XCMPLX_1:37
.=i by XCMPLX_1:27;
then hs2/.i=hs2.(len ((len gs1-'1) |-> j1)+(i+1-'len gs1)) by A109,
FINSEQ_4:def 4
.=gs2.(i+1-'len gs1) by A108,FINSEQ_1:def 7
.=gs2/.(i+1-'len gs1) by A108,FINSEQ_4:def 4;
(hs1/.i=hs1/.(i+1))& (hs2/.i,
end;
hence thesis by Def2;
end;
hence thesis by A11,A12,A13,A20,A26,A27;
end;

reserve S for set;

theorem
for Y being Subset of S,F being Matrix of n,m, bool S st
(ex i,j st i in Seg n & j in Seg m & F*(i,j) c= Y) &
(for i1,j1,i2,j2 st i1 in Seg n & i2 in Seg n & j1 in Seg m & j2 in Seg m
F*(i1,j1)c=Y iff F*(i2,j2)c=Y)
holds (for i,j st i in Seg n & j in Seg m holds F*(i,j)c=Y)
proof let Y be Subset of S,F be Matrix of n,m, (bool S);
assume A1:(ex i,j st i in Seg n & j in Seg m & F*(i,j) c= Y)&
(for i1,j1,i2,j2 st i1 in Seg n & i2 in Seg n & j1 in Seg m & j2 in Seg m
& i1,j1,i2,j2 are_adjacent2 holds F*(i1,j1)c=Y iff F*(i2,j2)c=Y);
then consider i1,j1 such that
A2:i1 in Seg n & j1 in Seg m & F*(i1,j1) c= Y;
A3:1<=i1 & i1<=n & 1<=j1 & j1<=m by A2,FINSEQ_1:3;
then i1-1>=1-1 & j1-1>=1-1 by REAL_1:49;
then A4:i1-'1=i1-1 & j1-'1=j1-1 by BINARITH:def 3;
thus for i,j st i in Seg n & j in Seg m holds F*(i,j)c=Y
proof let i2,j2;assume i2 in Seg n & j2 in Seg m;
then A5:1<=i2 & i2<=n & 1<=j2 & j2<=m by FINSEQ_1:3;
then i2-1>=1-1 & j2-1>=1-1 by REAL_1:49;
then A6:i2-'1=i2-1 & j2-'1=j2-1 by BINARITH:def 3;
now per cases;
case n=0 or m=0; hence contradiction by A2,FINSEQ_1:4;
case n<>0 & m<>0; then n>0 & m>0 by NAT_1:19;
then n>=0+1 & m>=0+1 by NAT_1:38;
then n-1>=0 & m-1>=0 by REAL_1:84;
then A7:n-'1=n-1 & m-'1=m-1 by BINARITH:def 3;
then A8:i1-'1<=n-'1 & j1-'1<=m-'1 by A3,A4,REAL_1:49;
i2-'1<=n-'1 & j2-'1<=m-'1 by A5,A6,A7,REAL_1:49;
then consider fs1,fs2 being FinSequence of NAT such that
A9:(for i,k1,k2 st i in dom fs1 & k1=fs1.i & k2=fs2.i
holds k1 <= n-'1 & k2 <= m-'1)&
fs1.1=i1-'1 & fs1.(len fs1)=i2-'1 &
fs2.1=j1-'1 & fs2.(len fs2)=j2-'1 & len fs1=len fs2 &
len fs1=((i1-'1)-'(i2-'1))+((i2-'1)-'(i1-'1))
+((j1-'1)-'(j2-'1))+((j2-'1)-'(j1-'1))+1 &
for i st 1<=i & i<len fs1 holds
A10:1<=len fs1 by A9,NAT_1:29;
deffunc F(Nat) = (fs1/.\$1)+1;
ex p being FinSequence of NAT st len p = len fs1 &
for j st j in Seg len fs1 holds p.j = F(j) from SeqLambdaD;
then consider p1 being FinSequence of NAT such that
A11: len p1 = len fs1
& for k st k in Seg len fs1 holds p1.k=(fs1/.k)+1;
deffunc F(Nat) = (fs2/.\$1)+1;
ex p being FinSequence of NAT st len p = len fs2
& for k st k in Seg len fs2 holds p.k=F(k) from SeqLambdaD;
then consider p2 being FinSequence of NAT such that
A12: len p2 = len fs2
& for k st k in Seg len fs2 holds p2.k=(fs2/.k)+1;
A13:1 in Seg len fs1 by A10,FINSEQ_1:3;
then A14:1 in dom fs1 by FINSEQ_1:def 3;
1-1<=len fs1-1 by A10,REAL_1:49;
then A15:len fs1 -'1=len fs1 -1 by BINARITH:def 3;
A16:1 in dom p1 by A11,A13,FINSEQ_1:def 3;
A17:fs1/.1=i1-'1 by A9,A14,FINSEQ_4:def 4;
A18: p1/.1=p1.1 by A16,FINSEQ_4:def 4 .=i1-'1 +1 by A11,A13,A17 .=
i1 by A4,XCMPLX_1:27;
A19:1 in dom fs2 by A9,A13,FINSEQ_1:def 3;
A20:1 in dom p2 by A9,A12,A13,FINSEQ_1:def 3;
defpred P[Nat] means
\$1+1<=len p1 implies F*(p1/.(\$1+1),p2/.(\$1+1)) c= Y;
A21:fs2/.1=j1-'1 by A9,A19,FINSEQ_4:def 4;
p2/.1=p2.1 by A20,FINSEQ_4:def 4 .=j1-'1 +1 by A9,A12,A13,A21 .=j1 by
A4,XCMPLX_1:27;
then A22: P[0] by A2,A18;
A23:  for k st P[k] holds P[k+1]
proof let k;assume A24:k+1<=len p1 implies
F*(p1/.(k+1),p2/.(k+1)) c= Y;
A25:1<=k+1 by NAT_1:37;
now per cases;
case A26: k+1<=len p1;
now per cases;
case A27:k+1+1<=len p1; then A28: k+1<len p1 by NAT_1:38;
then A29:k+1 in Seg len p1 by A25,FINSEQ_1:3;
then A30:k+1 in dom p1 by FINSEQ_1:def 3;
A31:k+1 in dom fs1 by A11,A29,FINSEQ_1:def 3;
set kp1=p1/.(k+1),kp2=p2/.(k+1);
set lp1=fs1/.(k+1),lp2=fs2/.(k+1);
A32:k+1 in dom p2 by A9,A11,A12,A29,FINSEQ_1:def 3;
A33:k+1 in dom fs2 by A9,A11,A29,FINSEQ_1:def 3;
A34:p1/.(k+1)=p1.(k+1) by A30,FINSEQ_4:def 4
.=lp1+1 by A11,A29;
A35:lp1=fs1.(k+1) & lp2=fs2.(k+1) by A31,A33,FINSEQ_4:def 4;
then lp1<=n-1 by A7,A9,A31;
then lp1+1<=n-1+1 by AXIOMS:24;
then A36:kp1<=n by A34,XCMPLX_1:27;
1<=1+lp1 by NAT_1:29;
then A37:p1/.(k+1) in Seg n by A34,A36,FINSEQ_1:3;
A38:p2/.(k+1)=p2.(k+1) by A32,FINSEQ_4:def 4
.=lp2+1 by A9,A11,A12,A29;
lp2<=m-'1 by A9,A31,A35;
then lp2+1<=m-1+1 by A7,AXIOMS:24;
then A39: kp2<=m by A38,XCMPLX_1:27;
1<=1+lp2 by NAT_1:29;
then A40:p2/.(k+1) in Seg m by A38,A39,FINSEQ_1:3;
1<=k+1+1 by NAT_1:37;
then A41:k+1+1 in Seg len p1 by A27,FINSEQ_1:3;
then A42: k+1+1 in dom p1 by FINSEQ_1:def 3;
set kp11=p1/.(k+1+1),kp21=p2/.(k+1+1);
set lp11=fs1/.(k+1+1),lp21=fs2/.(k+1+1);
A43:k+1+1 in dom fs1 by A11,A41,FINSEQ_1:def 3;
A44:k+1+1 in dom p2 by A9,A11,A12,A41,FINSEQ_1:def 3;
A45:k+1+1 in dom fs2 by A9,A11,A41,FINSEQ_1:def 3;
A46:p1/.(k+1+1)=p1.(k+1+1) by A42,FINSEQ_4:def 4
.=lp11+1 by A11,A41;
A47:fs1/.(k+1+1)=fs1.(k+1+1) by A43,FINSEQ_4:def 4;
A48: fs2/.(k+1+1)=fs2.(k+1+1) by A45,FINSEQ_4:def 4;
then lp11<=n-1 by A7,A9,A43,A47;
then A49: lp11+1<=n-1+1 by AXIOMS:24;
then A50:kp11<=n by A46,XCMPLX_1:27;
k+1+1 in dom p2 by A9,A11,A12,A41,FINSEQ_1:def 3;
then p2.(k+1+1)=p2/.(k+1+1) by FINSEQ_4:def 4;
then A51:p2/.(k+1+1)=lp21+1 by A9,A11,A12,A41;
lp21<=m-'1 by A9,A43,A47,A48;
then lp21+1<=m-1+1 by A7,AXIOMS:24;
then A52:kp11 <= n & kp21 <= m by A46,A49,A51,XCMPLX_1:27;
1<=1+lp11 by NAT_1:29;
then A53:p1/.(k+1+1) in Seg n by A46,A50,FINSEQ_1:3;
A54:p2/.(k+1+1)=p2.(k+1+1) by A44,FINSEQ_4:def 4
.=lp21+1 by A9,A11,A12,A41;
1<=1+lp21 by NAT_1:29;
then A55:p2/.(k+1+1) in Seg m by A52,A54,FINSEQ_1:3;
by A9,A11,A25,A28;
then p1/.(k+1),p2/.(k+1),p1/.(k+1+1),p2/.(k+1+1)
hence k+1+1<=len p1 implies F*(p1/.(k+1+1),p2/.(k+1+1)) c= Y
by A1,A24,A26,A37,A40,A53,A55;
case k+1+1>len p1;
hence k+1+1<=len p1 implies F*(p1/.(k+1+1),p2/.(k+1+1)) c= Y;
end;
hence k+1+1<=len p1 implies F*(p1/.(k+1+1),p2/.(k+1+1)) c= Y;
case k+1>len p1;
hence k+1+1<=len p1 implies F*(p1/.(k+1+1),p2/.(k+1+1)) c= Y
by NAT_1:38;
end;
hence k+1+1<=len p1 implies F*(p1/.(k+1+1),p2/.(k+1+1)) c= Y;
end;
A56:for i holds P[i] from Ind(A22,A23);
len fs1 -'1+1=len fs1 by A15,XCMPLX_1:27;
then A57:F*(p1/.len fs1,p2/.len fs1) c= Y by A11,A56;
A58:len fs1 in Seg len fs1 by A10,FINSEQ_1:3;
then A59:len fs1 in dom fs1 by FINSEQ_1:def 3;
A60:len fs1 in dom fs2 by A9,A58,FINSEQ_1:def 3;
A61:len fs1 in dom p1 by A11,A58,FINSEQ_1:def 3;
A62:len fs1 in dom p2 by A9,A12,A58,FINSEQ_1:def 3;
A63:(p1/.len fs1)=p1.(len fs1) by A61,FINSEQ_4:def 4
.= (fs1/.len fs1)+1 by A11,A58;
fs1/.len fs1=i2-'1 by A9,A59,FINSEQ_4:def 4;
then A64:(p1/.len fs1)=i2 by A6,A63,XCMPLX_1:27;
A65:(p2/.len fs1)=p2.(len fs1) by A62,FINSEQ_4:def 4
.= (fs2/.len fs1)+1 by A9,A12,A58;
fs2/.len fs1=j2-'1 by A9,A60,FINSEQ_4:def 4;
hence F*(i2,j2) c=Y by A6,A57,A64,A65,XCMPLX_1:27;
end;
hence F*(i2,j2) c=Y;
end;
end;
```