theorem
  p out a,b iff (Collinear a,p,b & not between a,p,b)
  proof
    p out a,b implies (Collinear a,p,b & not between a,p,b)
    proof
      assume
A1:   p out a,b;
      between p,b,a implies Collinear a,p,b & not between a,p,b
      proof
        assume between p,b,a;
        hence Collinear a,p,b;
        thus not between a,p,b
        proof
          assume
A2:       between a,p,b;
          then between b,p,a by Satz3p2;
          hence contradiction by A1,A2,Satz3p4;
        end;
      end;
      hence thesis by Satz3p2,Satz3p4,A1;
    end;
    hence thesis by Satz3p1,Satz3p2;
  end;
