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;
