theorem Satz3p15p4:
  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 being POINT of S st between4 a1,a2,a3,a4 &
  a1,a2,a3,a4 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 be POINT of S such that
A1: between a1,a2,a3 & a1,a2,a3 are_mutually_distinct by Satz3p15p3;
    consider a4 be POINT of S such that
A2: between a2,a3,a4 & a2,a3,a4 are_mutually_distinct by A1,Satz3p15p3;
    take a3,a4;
    thus between4 a1,a2,a3,a4 by A1,A2,Satz3p7p1,Satz3p7p2;
    thus a1,a2,a3,a4 are_mutually_distinct by A1,Satz3p7p1,GTARSKI1:def 10,A2;
  end;
