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
  A,A9 Is p implies ex E being Subset of S st E is_plane & A c= E &
  A9 c= E & Plane(A,A9) = E
  proof
    assume
A1: A,A9 Is p;
    then p in A /\ A9 by XBOOLE_0:def 4;
    then consider r be POINT of S such that
A2: not r in A and r in A9 and
A3: Plane(A,A9) = Plane(A,r) by A1,Def13;
    consider E be Subset of S such that
A4: E is_plane and A c= E and r in E and
A5: Plane(A,r) = E by A1,A2,Th50;
    take E;
    thus thesis by A1,A3,A4,A5,Th43;
  end;
