theorem Th12:
  a*(x - y) = a*x + -a*y & a*(x - y) = a*x + (-a)*y & a*(x - y) = a*x - a*y
proof
  thus
A1: a*(x - y) = a*x + a*(-y) by EUCLID_4:6
    .= a*x + -a*y by Th3;
  hence a*(x - y) = a*x + (-a)*y by Th3;
  thus thesis by A1;
end;
