theorem Th45:
  |{ p, q, p <X> q }| = |(q,q)| * |(p,p)| - |(q,p)| * |(p,q)|
  proof
    |{ p, q, p <X> q }| = |( p , q <X> (p <X> q) )| by EUCLID_5:def 5
                       .= |( p, |(q,q)| * p - |(q,p)| * q )| by EUCLID_5:32
                       .= |(p, |(q,q)| * p)| - |( p, |(q,p)| * q)|
                          by EUCLID_2:27
                       .= |(q,q)| * |(p,p)| - |( p, |(q,p)| * q)|
                          by EUCLID_2:20
                       .= |(q,q)| * |(p,p)| - |(q,p)| * |(p,q)| by EUCLID_2:20;
    hence thesis;
  end;
