
theorem Th12:
  for K being Field, a,b being Element of K, M being Matrix of K
  holds (a+b)*M=a*M + b*M
proof
  let K be Field, a,b be Element of K, M be Matrix of K;
A1: len (a*M)=len (M) & width (a*M)=width (M) by MATRIX_3:def 5;
A2: len ((a+b)*M)=len (M) & width ((a+b)*M)=width (M) by MATRIX_3:def 5;
A3: for i,j be Nat st [i,j] in Indices ((a+b)*M) holds ((a+b)*M)*(i,j)=(a*M
  + b*M)*(i,j)
  proof
    let i,j be Nat;
    assume
A4: [i,j] in Indices ((a+b)*M);
A5: Indices ((a+b)*M)=Indices M by A2,MATRIX_4:55;
    Indices (a*M)=Indices M by A1,MATRIX_4:55;
    then (a*M + b*M)*(i,j) =(a*M)*(i,j)+(b*M)*(i,j) by A4,A5,MATRIX_3:def 3
      .=a*(M*(i,j))+(b*M)*(i,j) by A4,A5,MATRIX_3:def 5
      .=a*(M*(i,j))+b*(M*(i,j)) by A4,A5,MATRIX_3:def 5
      .=(a+b)*(M*(i,j)) by VECTSP_1:def 7;
    hence thesis by A4,A5,MATRIX_3:def 5;
  end;
  len (a*M + b*M)=len (a*M) & width (a*M + b*M)=width (a*M) by MATRIX_3:def 3;
  hence thesis by A2,A1,A3,MATRIX_0:21;
end;
