theorem
  (-A) * B = -(A * B)
proof
  thus (-A) * B c= -(A * B)
  proof
    let x be object;
    assume x in (-A) * B;
    then consider a,b such that
A1: x = a * b and
A2: a in (-A) and
A3: b in B;
    consider c such that
A4: a = -c & c in A by A2;
    x = -(c * b) & c * b in A * B by A1,A3,A4,Th26;
    hence thesis;
  end;
  let x be object;
  assume x in -(A * B);
  then consider a such that
A5: x = (-a) and
A6: a in A * B;
  consider b,c such that
A7: a = b * c and
A8: b in A and
A9: c in B by A6;
A10: (-b) in (-A) by A8;
  x = (-b) * c by A5,A7,Th26;
  hence thesis by A9,A10;
end;
