reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th10:
  for M being Matrix of COMPLEX holds (-1)*M = -M
proof
  let M be Matrix of COMPLEX;
A1: width (-M) = width M by Th8;
A2: width ((-1)*M) = width M by Th2;
A3: len ((-1)*M) = len M by Th2;
A4: now
    let i,j;
    assume
A5: [i,j] in Indices ((-1)*M);
    then
A6: 1<= i by Th1;
A7: 1<=j by A5,Th1;
A8: j<=width M by A2,A5,Th1;
    i<=len M by A3,A5,Th1;
    then
A9: [i,j] in Indices M by A6,A7,A8,Th1;
    hence ((-1)*M)*(i,j) = (-1)*(M*(i,j)) by Th3
      .= -(M*(i,j))
      .= (-M)*(i,j) by A9,Th9;
  end;
  len (-M) = len M by Th8;
  hence thesis by A3,A1,A2,A4,MATRIX_0:21;
end;
