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 Th30:
  A is_line & not r in A implies half-plane(A,r) c= Plane(A,r)
  proof
    assume A is_line & not r in A;
    then consider r9 be POINT of S such that
    between r,A,r9 and
A1: Plane(A,r) = half-plane(A,r) \/ A \/ half-plane(A,r9) by Def10;
    half-plane(A,r) c= half-plane(A,r) \/ A &
      half-plane(A,r) \/ A c= half-plane(A,r) \/ A \/ half-plane(A,r9)
      by XBOOLE_1:7;
    hence thesis by A1;
  end;
