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,l,k) = ILine(M,k,l)
proof
A1: width ILine(M,k,l) = width M by Th1;
A2: for i,j st [i,j] in Indices (ILine(M,l,k)) holds ILine(M,l,k)*(i,j) =
  ILine(M,k,l)*(i,j)
  proof
    let i,j;
A3: Indices (ILine(M,l,k)) = Indices M by MATRIX_0:26;
    assume [i,j] in Indices (ILine(M,l,k));
    then
A4: i in dom M & j in Seg width M by A3,ZFMISC_1:87;
    then
A5: i = k implies ILine(M,l,k)*(i,j) = M*(l,j) by Def1;
A6: i = l implies ILine(M,k,l)*(i,j) = M*(k,j) by A4,Def1;
A7: i <> l & i <> k implies ILine(M,l,k)*(i,j) = M*(i,j) by A4,Def1;
A8: i = k implies ILine(M,k,l)*(i,j) = M*(l,j) by A4,Def1;
    i = l implies ILine(M,l,k)*(i,j) = M*(k,j) by A4,Def1;
    hence thesis by A4,A5,A7,A6,A8,Def1;
  end;
  len ILine(M,l,k) = len M & len ILine(M,k,l) = len M by Def1;
  hence thesis by A1,A2,Th1,MATRIX_0:21;
end;
