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 Th31:
  A is_line & not r in A implies A c= Plane(A,r) & r in Plane(A,r)
  proof
    assume that
A1: A is_line and
A2: not r in A;
    ex r9 be POINT of S st between r,A,r9 &
      Plane(A,r) = half-plane(A,r) \/ A \/ half-plane(A,r9) by A1,A2,Def10;
    then
A3: A c= half-plane(A,r) \/ A & half-plane(A,r) \/ A c= Plane(A,r)
      by XBOOLE_1:7;
    r in half-plane(A,r) & half-plane(A,r) c= Plane(A,r) by Th20,Th30,A1,A2;
    hence thesis by A3;
  end;
