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 Th40:
  A,A9 Is p implies A c= Plane(A9,A) & A9 c= Plane(A,A9)
  proof
    assume
A1: A,A9 Is p;
    then
A2: A is_line & A9 is_line & A <> A9 & A /\ A9 is non empty by XBOOLE_0:def 4;
    then consider r be POINT of S such that
A3: not r in A and
A4: r in A9 and
A5: Plane(A,A9) = Plane(A,r) by Def13;
    consider s be POINT of S such that
A6: not s in A9 and
A7: s in A and
A8: Plane(A9,A) = Plane(A9,s) by A2,Def13;
    s <> p & A9,A Is p by A6,A1;
    hence A c= Plane(A9,A) by A8,A7,Th37;
    thus A9 c= Plane(A,A9) by A1,A3,A4,A5,Th37;
end;
