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 Prelim11:
  a <> b & c <> c9 & (c in Line(a,b) or c in Line(b,a)) &
  (c9 in Line(a,b) or c9 in Line(b,a)) implies
  Line(c,c9) = Line(a,b) & Line(c,c9) = Line(b,a) &
  Line(c9,c) = Line(b,a) & Line(c9,c) = Line(a,b)
  proof
    assume that
A1: a <> b and
A2: c <> c9 and
A3: c in Line(a,b) or c in Line(b,a) and
A4: c9 in Line(a,b) or c9 in Line(b,a);
    Line(a,b) is_line by A1,GTARSKI3:def 11;
    hence thesis by A3,A4,A2,GTARSKI3:87;
  end;
