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

theorem Th6:
  for n,i,j st i<=n & j<=n
   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_adjacent
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 ) and
A2: for i1 st 1<=i1 & i1<len fs1 holds fs1.(i1+1)= (fs1/.i1)+1 or fs1.i1
  = (fs1/.(i1+1)) +1 by Th5;
  for i1 st 1<=i1 & i1<len fs1 holds fs1/.i1,fs1/.(i1+1) are_adjacent
  proof
    let i1;
    assume
A3: 1<=i1 & i1<len fs1;
    then
A4: fs1.i1= fs1/.i1 by FINSEQ_4:15;
    1<= i1+1 & i1+1<=len fs1 by A3,NAT_1:13;
    then
A5: fs1.(i1+1)= fs1/.(i1+1) by FINSEQ_4:15;
    fs1.(i1+1)= (fs1/.i1)+1 or fs1.i1= (fs1/.(i1+1)) +1 by A2,A3;
    hence thesis by A5,A4;
  end;
  hence thesis by A1;
end;
