reserve i,j for Nat;

theorem
  for K being non empty multMagma,
      a being Element of K, M being (Matrix of K),
      i being Nat st 1<=i & i<=len M holds
    Line(a*M,i)=a*Line(M,i)
proof
  let K be non empty multMagma, a be Element of K, M be Matrix of K;
  let i be Nat;
  assume
A1: 1<=i & i<=len M;
A2: width (a*M)=width M by MATRIX_3:def 5;
A3: Seg width M=Seg width (a*M) by MATRIX_3:def 5;
A4: len (a*Line(M,i))=len (Line(M,i)) & len (Line(M,i))=width M by Th16,
MATRIX_0:def 7;
  then
A5: dom (a*Line(M,i))=Seg width (a*M) by A2,FINSEQ_1:def 3;
  for j st j in Seg width (a*M) holds (a*Line(M,i)).j = (a*M)*(i,j)
  proof
    let j;
    assume
A6: j in Seg width (a*M);
    i in dom M by A1,FINSEQ_3:25;
    then [i,j] in Indices M by A3,A6,ZFMISC_1:87;
    then
A7: (a*M)*(i,j)=a*(M*(i,j)) by MATRIX_3:def 5;
    (Line(M,i)).j = M*(i,j) by A3,A6,MATRIX_0:def 7;
    hence thesis by A5,A6,A7,FVSUM_1:50;
  end;
  hence thesis by A4,A2,MATRIX_0:def 7;
end;
