reserve a,b for Real,
  i,j,n for Nat,
  M,M1,M2,M3,M4 for Matrix of n, REAL;

theorem
  |:a*M:|=|.a.|*|:M:|
proof
A1: len (a*M)=len M & width (a*M) =width M by MATRIXR1:27;
  len (|.a.|*|:M:|)=len |:M:| by MATRIXR1:27;
  then
A2: len (|.a.|*|:M:|)=len M by Def7;
A3: for i,j st [i,j] in Indices |:a*M:| holds |:a*M:|*(i,j) = (|.a.|*|:M:|)
  *(i,j)
  proof
A4: Indices (a*M) = Indices M by MATRIXR1:28;
A5: Indices (a*M) = Indices M by MATRIXR1:28;
    let i,j;
    assume
A6: [i,j] in Indices |:a*M:|;
A7: Indices |:M:| = Indices M by Th5;
A8: Indices |:a*M:| = Indices (a*M) by Th5;
    then
A9: |:a*M:|*(i,j)=|.(a*M)*(i,j).| by A6,Def7
      .=|.a*(M*(i,j)).| by A6,A8,A4,Th4
      .=|.a.|*|.M*(i,j).| by COMPLEX1:65;
    (|.a.|*|:M:|)*(i,j)=|.a.|*(|:M:|*(i,j)) by A6,A8,A5,Th4,A7
      .=|:a*M:|*(i,j) by A6,A8,A9,A5,Def7;
    hence thesis;
  end;
  width (|.a.|*|:M:|) =width |:M:| by MATRIXR1:27;
  then
A10: width (|.a.|*|:M:|) =width M by Def7;
  len |:a*M:|=len (a*M) & width |:a*M:|=width (a*M) by Def7;
  hence thesis by A1,A2,A10,A3,MATRIX_0:21;
end;
