reserve            S for satisfying_CongruenceSymmetry
                         satisfying_CongruenceEquivalenceRelation
                         TarskiGeometryStruct,
         a,b,c,d,e,f for POINT of S;
reserve S for satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        q,a,b,c,a9,b9,c9,x1,x2 for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve       S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d for POINT of S;
reserve         S for satisfying_CongruenceIdentity
                      satisfying_SegmentConstruction
                      satisfying_BetweennessIdentity
                      satisfying_Pasch
                      TarskiGeometryStruct,
        a,b,c,d,e for POINT of S;
reserve       S for satisfying_Tarski-model
                    TarskiGeometryStruct,
      a,b,c,d,p for POINT of S;
reserve                   S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9 for POINT of S;
reserve S for satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,a9,b9,c9,d9,p,q for POINT of S;
reserve                       S for satisfying_Tarski-model
                                    TarskiGeometryStruct,
        a,b,c,d,e,f,a9,b9,c9,d9 for POINT of S;
reserve p for POINT of S;
reserve r for POINT of S;
reserve x,y for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct;
reserve p,q,r,s for POINT of S;
reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
  a,b,p,q for POINT of S;
reserve S for non empty satisfying_Tarski-model TarskiGeometryStruct,
                  A,B for Subset of S,
        a,b,c,p,q,r,s for POINT of S;

theorem
  for S being non empty satisfying_Tarski-model TarskiGeometryStruct
  for a,b being POINT of S st
  S is satisfying_A8 &
  a <> b holds ex c being POINT of S st not Collinear a,b,c
  proof
    let S be non empty satisfying_Tarski-model TarskiGeometryStruct;
    let a,b be POINT of S;
    assume that
A1: S is satisfying_A8 and
A2: a <> b;
    assume
A3: for c be POINT of S holds Collinear a,b,c;
    consider a9,b9,c9 be POINT of S such that
A4: not Collinear a9,b9,c9 by A1,Satz6p24;
A5: a9 <> b9 by A4,Satz3p1;
    set A = Line(a,b);
    Collinear a,b,a9 & Collinear a,b,b9 by A3;
    then A is_line & a9 in A & b9 in A by A2; then
A6: Line(a9,b9) = A by A5,Satz6p18;
    Collinear a,b,c9 by A3;
    then c9 in Line(a9,b9) by A6;
    then ex x be POINT of S st c9 = x & Collinear a9,b9,x;
    hence contradiction by A4;
  end;
