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 Prelim08b: ::: Lemma ext1, Chap. 8A
  for S being non empty satisfying_Tarski-model TarskiGeometryStruct
  for a,b,c,d being POINT of S st a <> b & Collinear a,b,c &
  Collinear a,b,d holds Collinear a,c,d
  proof
    let S be non empty satisfying_Tarski-model TarskiGeometryStruct;
    let a,b,c,d be POINT of S;
    assume that
A1: a <> b and
A2: Collinear a,b,c and
A3: Collinear a,b,d;
    per cases;
    suppose a = c;
      hence thesis by Prelim05;
    end;
    suppose a <> c; then
A4:   Line(a,b) = Line(a,c) by A1,A2,Prelim07;
      d in {y where y is POINT of S: Collinear a,b,y} by A3;
      then d in Line(a,c) by A4,GTARSKI3:def 10; then
      Collinear d,a,c by LemmaA2;
      hence thesis by GTARSKI3:45;
    end;
  end;
