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 Prelim12:
  Middle reflection(p,c),reflection(p,b),reflection(p,reflection(b,c))
  proof
    Middle c,b,reflection(b,c) by GTARSKI3:def 13;
    then between reflection(p,c),reflection(p,b),reflection(p,reflection(b,c))
    & reflection(p,b),reflection(p,c) equiv
    reflection(p,b),reflection(p,reflection(b,c)) by GTARSKI3:def 12,106,107;
    hence thesis by GTARSKI3:def 12;
  end;
