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 is_line implies for E1,E2 being Subset of S st E1 is_plane &
  E2 is_plane & A c= E1 & A c= E2 & E1 <> E2 holds (for x being POINT of S
  holds (x in E1 & x in E2) iff x in A)
  proof
    assume that
A1: A is_line;
    let E1,E2 be Subset of S;
    assume that
A2: E1 is_plane and
A3: E2 is_plane and
A4: A c= E1 and
A5: A c= E2 and
A6: E1 <> E2;
    consider a,b be POINT of S such that a <> b and
A7: A = Line(a,b) by A1;
    let x be POINT of S;
    hereby
      assume that
A8:   x in E1 and
A9:   x in E2;
      per cases;
      suppose x <> a & x <> b;
        per cases;
        suppose x in A;
          hence x in A;
        end;
        suppose
A10:      not x in A;
          not Collinear a,b,x & a in E1 & b in E1 & a in E2 & b in E2
            by A4,A5,A7,A10,GTARSKI3:83;
          hence x in A by A2,A3,A6,A8,A9,Th52;
        end;
      end;
      suppose x = a or x = b;
        hence x in A by A7,GTARSKI3:83;
      end;
    end;
    assume x in A;
    hence x in E1 & x in E2 by A4,A5;
  end;
