theorem Satz7p15:
  between p,q,r iff between reflection(a,p),reflection(a,q),reflection(a,r)
  proof
    now
      assume C1: between reflection(a,p),reflection(a,q),reflection(a,r);
      reflection(a,reflection(a,p)) = p & reflection(a,reflection(a,q)) = q &
        reflection(a,reflection(a,r)) = r by Satz7p7;
      hence between p,q,r by C1,Lemma7p15a;
    end;
    hence thesis by Lemma7p15a;
  end;
