
theorem
  for K being Field, a,b being Element of K, M being Matrix of K holds a
  *(b*M)=(a*b)*M
proof
  let K be Field, a,b be Element of K, M being Matrix of K;
A1: len ((a*b)*M)=len M & width ((a*b)*M)=width M by MATRIX_3:def 5;
A2: len (a*(b*M))=len (b*M) by MATRIX_3:def 5;
A3: width (a*(b*M))=width (b*M) by MATRIX_3:def 5;
  then
A4: width (a*(b*M)) =width M by MATRIX_3:def 5;
A5: len (b*M)=len M & width (b*M)=width M by MATRIX_3:def 5;
A6: for i,j being Nat st [i,j] in Indices (a*(b*M)) holds (a*(b*M))*(i,j)=((
  a*b)*M)*(i,j)
  proof
    let i,j be Nat;
    assume
A7: [i,j] in Indices (a*(b*M));
A8: Indices ((b*M))=Indices (M) by A5,MATRIX_4:55;
A9: Indices (a*(b*M))=Indices ((b*M)) by A2,A3,MATRIX_4:55;
    then (a*(b*M))*(i,j) = (a*((b*M)*(i,j))) by A7,MATRIX_3:def 5
      .= (a*(b*(M*(i,j)))) by A7,A9,A8,MATRIX_3:def 5
      .= (a*b)*(M*(i,j)) by GROUP_1:def 3;
    hence thesis by A7,A9,A8,MATRIX_3:def 5;
  end;
  len (a*(b*M)) =len M by A2,MATRIX_3:def 5;
  hence thesis by A1,A4,A6,MATRIX_0:21;
end;
