reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,c9,x,y,z,p,q,q9 for POINT of S;
reserve              S for satisfying_Tarski-model TarskiGeometryStruct,
        a,a9,b,b9,c,c9 for POINT of S;
reserve S                 for non empty satisfying_Tarski-model
                                    TarskiGeometryStruct,
        A,A9              for Subset of S,
        x,y,z,a,b,c,c9,d,u,p,q,q9 for POINT of S;

theorem Satz8p18Lemma:
  not Collinear a,b,c & between b,a,y & a <> y &
    between a,y,z & y,z equiv y,p & y <> p &
    q9 = reflection(z,q) & Middle c,x,c9 & c <> y &
    between q9,y,c9 & Middle y,p,c & between p,y,q & q <> q9
  implies x <> y
  proof
    assume that
A1: not Collinear a,b,c and
A2: between b,a,y and
A3: a <> y and
A4: between a,y,z and
A5: y,z equiv y,p and
A6: y <> p and
A7: q9 = reflection(z,q) and
A8: Middle c,x,c9 and
A9: c <> y and
A10: between q9,y,c9 and
A11: Middle y,p,c and
A12: between p,y,q and
A13: q <> q9;
    assume
A14: x = y;
    now
      Collinear b,a,y by A2,GTARSKI1:def 17;
      hence y in Line(a,b) by LemmaA1;
      thus a <> y by A3;
      thus a <> b by A1,GTARSKI3:46;
    end;
    then
A15: Line(a,y) = Line(a,b) by GTARSKI3:82;
    now
      Collinear a,y,z by A4,GTARSKI1:def 17;
      hence z in Line(y,a) by LemmaA1;
      thus y <> a by A3;
      thus y <> z by A5,A6,GTARSKI1:def 7,GTARSKI3:4;
    end; then
A16: Line(y,z) = Line(a,b) by A15,GTARSKI3:82;
A17: Line(x,c) = Line(x,c9) by A8,A9,A14,Prelim10;
    Collinear x,c9,q9 by A10,A14,GTARSKI1:def 17;
    then
A18: q9 in Line(x,c) by A17,LemmaA1;
    between c,p,x & between p,x,q & p <> x
      by A14,A6,A11,A12,GTARSKI3:def 12,14;
    then between c,x,q by GTARSKI3:19;
    then Collinear c,x,q by GTARSKI1:def 17;
    then q in Line(c,x) by LemmaA1;
    then
A19: Line(q,q9) = Line(c,x) by A13,A9,A14,A18,Prelim11;
    Middle q,z,q9 by A7,GTARSKI3:def 13;
    then between q,z,q9 by GTARSKI3:def 12;
    then Collinear q,z,q9 by GTARSKI1:def 17;
    then Collinear q,q9,z by GTARSKI3:45;
    then z in Line(c,x) by A19,LemmaA1; then
    Collinear z,c,x by LemmaA2;
    then Collinear y,z,c by A14,GTARSKI3:45;
    then c in Line(a,b) by A16,LemmaA1; then
    Collinear c,a,b by LemmaA2;
    hence contradiction by A1,GTARSKI3:45;
  end;
