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 Th22:
  b in half-plane(A,a) implies half-plane(A,b) c= half-plane(A,a)
  proof
    assume b in half-plane(A,a);
    then consider t be POINT of S such that
A1: b = t and
A2: A out t,a;
    half-plane(A,b) c= half-plane(A,a)
    proof
      let x be object;
      assume x in half-plane(A,b);
      then consider y be POINT of S such that
A3:   x = y and
A4:   A out y,b;
      A out y,a by A1,A2,A4,Th19;
      hence thesis by A3;
    end;
    hence thesis;
  end;
