theorem Satz7p4existence:
  for p being POINT of S holds ex p9 being POINT of S st Middle p,a,p9
  proof
    let p be POINT of S;
    per cases;
    suppose p <> a;
      consider x be POINT of S such that
A1:   between p,a,x & a,x equiv p,a by GTARSKI1:def 8;
      a,x equiv a,p by A1,Satz2p5;
      then a,p equiv a,x by Satz2p2;
      hence thesis by A1,DEFM;
    end;
    suppose p = a;
      hence thesis by Satz7p3;
    end;
  end;
