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 Satz3p15p3:
  for S being satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Lower_Dimension_Axiom
              TarskiGeometryStruct holds
  for a1,a2 being POINT of S st a1 <> a2 holds
  ex a3 being POINT of S st between a1,a2,a3 & a1,a2,a3 are_mutually_distinct
  proof
    let S be satisfying_CongruenceSymmetry
             satisfying_CongruenceEquivalenceRelation
             satisfying_CongruenceIdentity
             satisfying_SegmentConstruction
             satisfying_BetweennessIdentity
             satisfying_Lower_Dimension_Axiom
             TarskiGeometryStruct;
    let a1,a2 be POINT of S;
    assume
A1: a1 <> a2;
    consider a3 be POINT of S such that
A2: between a1,a2,a3 & a2 <> a3 by Satz3p14;
    a1 <> a3 by A2,GTARSKI1:def 10;
    hence thesis by A1,A2,ZFMISC_1:def 5;
  end;
