reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,c9,x,y,z,p,q,q9 for POINT of S;

theorem Prelim09:
  for S being non empty satisfying_Tarski-model TarskiGeometryStruct
  for a,b being POINT of S st Middle a,a,b or Middle a,b,b or
  Middle a,b,a holds a = b
  proof
    let S be non empty satisfying_Tarski-model TarskiGeometryStruct;
    let a,b be POINT of S;
    assume Middle a,a,b or Middle a,b,b or Middle a,b,a;
    then a,a equiv a,b or b,a equiv b,b or Middle a,b,a by GTARSKI3:def 12;
    hence thesis by GTARSKI1:def 7,GTARSKI3:3,GTARSKI3:97;
  end;
