reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th1:
  for M be tabular FinSequence holds [i,j] in Indices M iff 1<= i &
  i<=len M & 1<=j & j<=width M
proof
  let M be tabular FinSequence;
  thus [i,j] in Indices M implies 1<= i & i<=len M & 1<=j & j<=width M
  proof
    assume
A1: [i,j] in Indices M;
    then [i,j] in [:Seg len M,Seg width M:] by FINSEQ_1:def 3;
    then
A2: i in Seg len M by ZFMISC_1:87;
    j in Seg width M by A1,ZFMISC_1:87;
    hence thesis by A2,FINSEQ_1:1;
  end;
  assume that
A3: 1<= i and
A4: i<=len M and
A5: 1<=j and
A6: j<=width M;
A7: j in Seg width M by A5,A6,FINSEQ_1:1;
  i in dom M by A3,A4,FINSEQ_3:25;
  hence thesis by A7,ZFMISC_1:87;
end;
