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 Th11:
  l in dom M & k in dom M implies ILine(ILine(M,l,k),l,k) = M
proof
  assume
A1: l in dom M & k in dom M;
  set M1 = ILine(M,l,k);
A2: dom M1 = Seg len ILine(M,l,k) by FINSEQ_1:def 3
    .= Seg len M by Def1
    .= dom M by FINSEQ_1:def 3;
A3: width M1 = width M by Th1;
A4: for i,j st [i,j] in Indices M holds ILine(M1,l,k)*(i,j) = M*(i,j)
  proof
    let i,j;
    assume
A5: [i,j] in Indices M;
    then
A6: j in Seg width M by ZFMISC_1:87;
A7: i in dom M by A5,ZFMISC_1:87;
    then i <> l & i <> k implies ILine(M1,l,k)*(i,j) = M1*(i,j) by A3,A2,A6
,Def1;
    then
A8: i <> l & i <> k implies ILine(M1,l,k)*(i,j) = M*(i,j) by A7,A6,Def1;
    ILine(M1,l,k)*(l,j) = M1*(k,j) & ILine(M1,l,k)*(k,j) = M1*(l,j) by A1,A3,A2
,A6,Def1;
    hence thesis by A1,A6,A8,Def1;
  end;
A9: width ILine(M1,l,k) =width M1 by Th1;
  len M1 =len M & len ILine(M1,l,k) =len M1 by Def1;
  hence thesis by A9,A4,Th1,MATRIX_0:21;
end;
