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;
