reserve
  j, k, l, n, m, t,i for Nat,
  K for comRing, 
  a for Element of K,
  M,M1,M2 for Matrix of n,m,K,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem
  ILine(M,k,k) = M
proof
A1: for i,j st [i,j] in Indices M holds ILine(M,k,k)*(i,j) = M*(i,j)
  proof
    let i,j;
    assume [i,j] in Indices M;
    then
A2: i in dom M & j in Seg width M by ZFMISC_1:87;
    then i = k implies ILine(M,k,k)*(i,j) = M*(k,j) by Def1;
    hence thesis by A2,Def1;
  end;
  len ILine(M,k,k) = len M & width ILine(M,k,k) = width M by Def1,Th1;
  hence thesis by A1,MATRIX_0:21;
end;
