theorem
  a + b = k*a + k*b & a*b > 0 implies k = 1
  proof
    assume
    A0: a + b = k*a + k*b & a*b > 0; then
    (k-1)*(a + b) = 0; then
    a + b = 0 or k -1 = 0; then
    a = 0 or k = 1;
    hence thesis by A0;
  end;
