reserve x,y,z,x1,x2,y1,y2,z1,z2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Ring;
reserve a for Element of K;

theorem Lm100:
  for K being commutative non empty multMagma
  for M being Matrix of K
  for a being Element of K
  holds a*M = M*a
proof
  let K be commutative non empty multMagma;
  let M be Matrix of K;
  let a be Element of K;
  A1: len (a*M) = len M by Def5
    .= len (M*a) by Def6;
  A2: width (a*M) = width M by Def5
    .= width (M*a) by Def6;
  for i,j being Nat st [i,j] in Indices (a*M)
  holds (a*M) * (i,j) = (M*a) * (i,j)
  proof
    let i,j be Nat;
    assume A3: [i,j] in Indices (a*M);
    A4: [i,j] in Indices M
    proof
      A5: dom (a*M) = Seg (len (a*M)) by FINSEQ_1:def 3
        .= Seg (len M) by Def5
        .= dom M by FINSEQ_1:def 3;
      width (a*M) = width M by Def5;
      then [:dom(M),Seg(width M):] = [:dom(a*M),Seg(width(a*M)):] by A5
        .= Indices (a*M);
      then Indices M = Indices(a*M);
      hence thesis by A3;
    end;
    thus (a*M) * (i,j) = a * (M*(i,j)) by A4,Def5
      .= (M*a) * (i,j) by A4,Def6;
  end;
  hence a*M = M*a by A1,A2,MATRIX_0:21;
end;
