reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th9:
  for A being Matrix of REAL holds (-1)*A = -A
proof
  let A be Matrix of REAL;
A1: width ((-1)*A) = width A by Th5;
A2: len (((-1) qua Real)*A) = len A by Th5;
A3: now
    let i,j be Nat;
    reconsider i0=i,j0=j as Nat;
    assume
A4: [i,j] in Indices ((-1)*A);
    then
A5: 1<=j0 & j0<=width A by A1,MATRIXC1:1;
    1<= i0 & i0<=len A by A2,A4,MATRIXC1:1;
    then
A6: [i0,j0] in Indices A by A5,MATRIXC1:1;
    hence ((-1)*A)*(i,j) = (-1)*(A*(i0,j0)) by Th3
      .= -(A*(i,j))
      .= (-A)*(i,j) by A6,Lm3;
  end;
  len (-A) = len A & width (-A) = width A by MATRIX_3:def 2;
  hence thesis by A2,A1,A3,MATRIX_0:21;
end;
