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 Prelim08a:
  for S being satisfying_CongruenceIdentity
    satisfying_SegmentConstruction satisfying_BetweennessIdentity
    satisfying_Pasch TarskiGeometryStruct
  for a,b,c being POINT of S st
  (Middle a,b,c or between a,b,c) holds
  Collinear a,b,c & Collinear b,c,a & Collinear c,a,b &
  Collinear c,b,a & Collinear b,a,c & Collinear a,c,b
  proof
    let S be satisfying_CongruenceIdentity
      satisfying_SegmentConstruction satisfying_BetweennessIdentity
      satisfying_Pasch TarskiGeometryStruct;
    let a,b,c be POINT of S;
    assume Middle a,b,c or between a,b,c;
    then Collinear a,b,c by Prelim08;
    hence thesis by GTARSKI3:45;
  end;
