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;

theorem Satz3p17:
  for S being satisfying_Tarski-model
              TarskiGeometryStruct
  for a,b,c,p,a9,b9,c9 being POINT of S st between a,b,c & between a9,b9,c &
  between a,p,a9 holds ex q being POINT of S st between p,q,c & between b,q,b9
  proof
    let S be satisfying_Tarski-model TarskiGeometryStruct;
    let a,b,c,p,a9,b9,c9 be POINT of S;
    assume
A1: between a,b,c & between a9,b9,c & between a,p,a9;
    then between c,b9,a9 by Satz3p2;
    then consider x be POINT of S such that
A2: between b9,x,a & between p,x,c by A1,GTARSKI1:def 11;
    between c,b,a by A1,Satz3p2;
    then consider q be POINT of S such that
A3: between x,q,c & between b,q,b9 by A2,GTARSKI1:def 11;
    between p,q,c by A2,A3,Satz3p5p2;
    hence thesis by A3;
  end;
