
theorem Th20:
  for K being Field, b being Element of K,M1,M2 being Matrix of K
  st len M1 = len M2 & width M1 = width M2 holds b*(M1+M2)=b*M1 + b*M2
proof
  let K be Field, b be Element of K, M1,M2 be Matrix of K;
A1: len (b*(M1+M2))=len (M1+M2) & width (b*(M1+M2)) = width (M1+M2) by
MATRIX_3:def 5;
A2: len (M1+M2)=len M1 & width (M1+M2)=width M1 by MATRIX_3:def 3;
A3: len (b*M1)=len M1 & width (b*M1)=width M1 by MATRIX_3:def 5;
  assume
A4: len M1= len M2 & width M1=width M2;
A5: for i,j being Nat st [i,j] in Indices (b*(M1+M2)) holds (b*(M1+M2))*(i,j
  )=(b*M1 + b*M2)*(i,j)
  proof
    let i,j be Nat;
    assume
A6: [i,j] in Indices (b*(M1+M2));
A7: Indices M2=Indices M1 by A4,MATRIX_4:55;
A8: Indices (b*(M1+M2))=Indices (M1+M2) by A1,MATRIX_4:55;
A9: Indices (M1+M2)=Indices M1 by A2,MATRIX_4:55;
    Indices (b*M1)=Indices M1 by A3,MATRIX_4:55;
    then (b*M1 + b*M2)*(i,j) =(b*M1)*(i,j)+(b*M2)*(i,j) by A6,A8,A9,
MATRIX_3:def 3
      .=b*(M1*(i,j))+(b*M2)*(i,j) by A6,A8,A9,MATRIX_3:def 5
      .=b*(M1*(i,j))+b*(M2*(i,j)) by A6,A8,A9,A7,MATRIX_3:def 5
      .=(b*(M1*(i,j)+M2*(i,j))) by VECTSP_1:def 7;
    then (b*M1 + b*M2)*(i,j) = (b*((M1 + M2)*(i,j))) by A6,A8,A9,MATRIX_3:def 3
      .= (b*(M1 + M2)*(i,j)) by A6,A8,MATRIX_3:def 5;
    hence thesis;
  end;
  len (b*M1 + b*M2)=len (b*M1) & width (b*M1 + b*M2)=width (b*M1) by
MATRIX_3:def 3;
  hence thesis by A1,A2,A3,A5,MATRIX_0:21;
end;
