reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct;
reserve a,b for POINT of S;
reserve A for Subset of S;
reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct;
reserve a,b,c,m,r,s for POINT of S;
reserve A for Subset of S;
reserve S         for non empty satisfying_Lower_Dimension_Axiom
                                satisfying_Tarski-model
                                TarskiGeometryStruct,
        a,b,c,d,m,p,q,r,s,x for POINT of S,
        A,A9,E              for Subset of S;

theorem
  for S being satisfying_CongruenceIdentity
  satisfying_SegmentConstruction satisfying_BetweennessIdentity
  satisfying_Pasch satisfying_Lower_Dimension_Axiom
  TarskiGeometryStruct holds ex p,q being POINT of S st p <> q
  proof
    let S be satisfying_CongruenceIdentity
    satisfying_SegmentConstruction satisfying_BetweennessIdentity
    satisfying_Pasch satisfying_Lower_Dimension_Axiom
    TarskiGeometryStruct;
    assume
A1: not ex p,q being POINT of S st p <> q;
    consider a,b,c being POINT of S such that
A2: not between a,b,c and not between b,c,a and not between c,a,b
      by GTARSKI2:def 7;
    a = b & a = c by A1;
    hence contradiction by A2,GTARSKI3:15;
  end;
