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
  p,q out s,r implies s <> p & s <> q & r <> p & r <> q & p <> q
  proof
    assume
A1: p,q out s,r;
    then p <> q & Line(p,q) out s,r;
    then ex x be POINT of S st between s,Line(p,q),x & between r,Line(p,q),x;
    hence thesis by A1,GTARSKI3:83;
  end;
