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 Prelim07:
  for S being non empty satisfying_Tarski-model TarskiGeometryStruct
  for p,q,r being POINT of S st p <> q & p <> r &
  (Collinear p,q,r or Collinear q,r,p or Collinear r,p,q or
  Collinear p,r,q or Collinear q,p,r or Collinear r,q,p ) holds
  Line(p,q) = Line(p,r) & Line(p,q) = Line(r,p) &
  Line(q,p) = Line(p,r) & Line(q,p) = Line(r,p)
  proof
    let S be non empty satisfying_Tarski-model TarskiGeometryStruct;
    let p,q,r be POINT of S;
    assume that
A1: p <> q and
A2: p <> r and
A3: Collinear p,q,r or Collinear q,r,p or Collinear r,p,q or
      Collinear p,r,q or Collinear q,p,r or Collinear r,q,p;
    Collinear p,q,r by A3,GTARSKI3:45;
    then r on_line p,q by A1,GTARSKI3:84;
    hence thesis by A1,A2,GTARSKI1:39,GTARSKI3:85;
  end;
