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
  for S being satisfying_Tarski-model
              satisfying_Lower_Dimension_Axiom
              TarskiGeometryStruct holds
  for a1,a2 being POINT of S st a1 <> a2 holds
  ex a3,a4,a5 being POINT of S st between5 a1,a2,a3,a4,a5 &
  a1,a2,a3,a4,a5 are_mutually_distinct
  proof
    let S be satisfying_Tarski-model
             satisfying_Lower_Dimension_Axiom
             TarskiGeometryStruct;
    let a1,a2 be POINT of S;
    assume a1 <> a2;
    then consider a3,a4 be POINT of S such that
A1: between4 a1,a2,a3,a4 & a1,a2,a3,a4 are_mutually_distinct by Satz3p15p4;
    consider a5 be POINT of S such that
A2: between a3,a4,a5 & a3,a4,a5 are_mutually_distinct by A1,Satz3p15p3;
    take a3,a4,a5;
    between a1,a3,a5 & between a1,a4,a5 & between a2,a3,a5 & between a2,a4,a5
      by A1,A2,Satz3p7p1,Satz3p7p2;
    hence between5 a1,a2,a3,a4,a5 by A1,A2,Satz3p6p2;
    thus a1,a2,a3,a4,a5 are_mutually_distinct
      by A1,A2,Satz3p7p1,GTARSKI1:def 10;
  end;
