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;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct,
        a,b,m for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              TarskiGeometryStruct,
        a,b,m for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        a for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_SAS
              TarskiGeometryStruct,
  a,p,p9 for POINT of S;
reserve S for satisfying_CongruenceIdentity
              satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_SAS
              satisfying_Pasch
              TarskiGeometryStruct,
  a,p,p9 for POINT of S;
reserve S for satisfying_CongruenceIdentity
                satisfying_CongruenceSymmetry
                satisfying_CongruenceEquivalenceRelation
                satisfying_SegmentConstruction
                satisfying_BetweennessIdentity
                satisfying_SAS
                TarskiGeometryStruct,
        a,p for POINT of S;
reserve              S for satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,m,p,p9,q,r,s for POINT of S;
reserve         S for non empty satisfying_Tarski-model TarskiGeometryStruct,
        a,b,c,d,p for POINT of S;

theorem
  not Collinear a,b,c & b <> d & a,b equiv c,d & b,c equiv d,a &
  Collinear a,p,c & Collinear b,p,d implies Middle a,p,c & Middle b,p,d
  proof
    assume that
A1: not Collinear a,b,c and
A2: b <> d and
A3: a,b equiv c,d and
A4: b,c equiv d,a and
A5: Collinear a,p,c and
A6: Collinear b,p,d;
A7: Collinear b,d,p by A6,Satz3p2;
    then consider p9 being POINT of S such that
A8: b,d,p cong d,b,p9 by GTARSKI1:def 5,Satz4p14;
A9: Collinear d,b,p9 by A7,A8,Satz4p13;
    now
      thus Collinear b,d,p by A6,Satz3p2;
      thus b,d,p cong d,b,p9 by A8;
      a,b equiv d,c by A3,Satz2p5;
      hence b,a equiv d,c by Satz2p4;
      thus d,a equiv b,c by A4,Satz2p2;
    end;
    then
A10: b,d,p,a FS d,b,p9,c;
    then
    p,a equiv c,p9 by A2,Satz4p16,Satz2p5;
    then
A11: a,p equiv c,p9 by Satz2p4;
A12: a,c equiv c,a by GTARSKI1:def 5;
    now
      thus Collinear b,d,p by A6,Satz3p2;
      thus b,d,p cong d,b,p9 by A8;
      thus b,c equiv d,a by A4;
      a,b equiv d,c by A3,Satz2p5;
      then b,a equiv d,c by Satz2p4;
      hence d,c equiv b,a by Satz2p2;
    end;
    then b,d,p,c FS d,b,p9,a;
    then p,c equiv p9,a by A2,Satz4p16;
    then
A13: Collinear c,p9,a by A5,Satz4p13,A11,A12,GTARSKI1:def 3;
A14: a <> c by A1,Satz3p1;
A15: Line(a,c) <> Line(b,d)
    proof
      assume Line(a,c) = Line(b,d);
      then b in Line(a,c) by Satz6p17;
      then ex x be POINT of S st b = x & Collinear a,c,x;
      hence contradiction by A1,Satz3p2;
    end;
    now
      thus Line(a,c) <> Line(b,d) by A15;
      thus Line(a,c) is_line by A14;
      thus Line(b,d) is_line by A2;
      Collinear a,c,p9 by A13;
      hence p9 in Line(a,c);
      Collinear b,d,p9 by A9,Satz3p2;
      hence p9 in Line(b,d);
      Collinear a,c,p by A5,Satz3p2;
      hence p in Line(a,c);
      Collinear b,d,p by A6,Satz3p2;
      hence p in Line(b,d);
    end;
    then
A16: p9 = p by Satz6p19;
    now
      thus Middle a,p,c by A14,Satz7p20,A10,A2,Satz4p16,A16,A5;
      b,p equiv p,d by A16,A8,Satz2p5;
      hence Middle b,p,d by A2,Satz7p20,A6,Satz2p4;
    end;
    hence thesis;
  end;
