theorem Satz6p13:
  p out a,b implies (p,a <= p,b iff between p,a,b)
  proof
    assume
A1:  p out a,b;
A2:  p,a <= p,b implies between p,a,b
    proof
      assume
A3:   p,a <= p,b;
      consider y be POINT of S such that
A4:   between p,y,b & p,a equiv p,y by A3;
A5:   p out y,b by A1,A4,GTARSKI1:def 7;
      p,y equiv p,y by Satz2p1;
      hence thesis by A4,A1,A5,Satz6p11pb;
    end;
    between p,a,b implies p,a <= p,b
    proof
      assume
A6:   between p,a,b;
      then Collinear p,a,b;
      hence thesis by A6,Satz5p12;
    end;
    hence thesis by A2;
  end;
