theorem Th12:
  for M being tabular FinSequence holds [i,j] in Indices M iff i
  in Seg len M & j in Seg width M
proof
  let M be tabular FinSequence;
  hereby
    assume [i,j] in Indices M;
    then
A1: [i,j] in [:dom M,Seg width M:] by MATRIX_0:def 4;
    then i in dom M by ZFMISC_1:87;
    hence i in Seg len M & j in Seg width M by A1,FINSEQ_1:def 3,ZFMISC_1:87;
  end;
  assume that
A2: i in Seg len M and
A3: j in Seg width M;
  i in dom M by A2,FINSEQ_1:def 3;
  then [i,j] in [:dom M,Seg width M:] by A3,ZFMISC_1:87;
  hence thesis by MATRIX_0:def 4;
end;
