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 Th3:
  for K being non empty multMagma, a being Element of K, M being
  Matrix of n,m,K
  for i,j st i in dom M & j in Seg width M holds (a*Line(M,i)).
  j = a*(M*(i,j))
proof
  let K be non empty multMagma;
  let a be Element of K;
  let M be Matrix of n,m,K;
  let i,j;
  assume that
A1: i in dom M and
A2: j in Seg width M;
A3: [i,j] in Indices M by A1,A2,ZFMISC_1:87;
A4: j in Seg width (a * M) by A2,MATRIX_3:def 5;
  dom M = Seg len M by FINSEQ_1:def 3;
  then 1<=i & i<=len M by A1,FINSEQ_1:1;
  then (a*Line(M,i)).j = (Line(a*M,i)).j by MATRIXR1:20
    .= (a*M)*(i,j) by A4,MATRIX_0:def 7
    .= a*(M*(i,j)) by A3,MATRIX_3:def 5;
  hence thesis;
end;
