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 Th42:
  A,A9 Is p implies Plane(A,A9) c= Plane(A9,A)
  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;
    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) and
A6: A9 = Line(r,p) and
A7: ex r9 being POINT of S st (between r,p,r9 & p <> r9 & Collinear r,p,r9 &
      not r9 in A & Plane(A,r) = Plane(A,r9)) by A1,Th41;
    consider r9 be POINT of S such that
A8: between r,p,r9 and
    p <> r9 and
    Collinear r,p,r9 and
A9: not r9 in A and
A10: Plane(A,r) = Plane(A,r9) by A7;
A11: Collinear r,p,r9 by A8;
    then
A12: r9 in {x where x is POINT of S : Collinear r,p,x};
    now
      let x be object;
      assume
A13:  x in Plane(A,A9);
      Plane(A,r9) = {x where x is POINT of S : A out x,r9 or x in A or
        between r9,A,x} by A9,A1,Th32;
      then consider s be POINT of S such that
A14:  x = s and
A15:  A out s,r9 or s in A or between r9,A,s by A13,A5,A10;
      consider u be POINT of S such that
A16:  not u in A9 and u in A and
A17:  Plane(A9,A) = Plane(A9,u) by A2,Def13;
      per cases;
      suppose
A18:    s in A9;
        A9 c= Plane(A9,u) by A1,A16,Th31;
        hence x in Plane(A9,A) by A18,A14,A17;
      end;
      suppose
A19:    not s in A9;
        per cases by A15;
        suppose
A20:      s in A;
          A c= Plane(A9,A) by A1,Th40;
          hence x in Plane(A9,A) by A14,A20;
        end;
        suppose A out s,r9;
          then
A21:      A out r9,s;
          between r9,A,r by A1,A3,A9,A8,GTARSKI3:14;
          then between s,A,r by A21,Th14;
          then consider t be POINT of S such that
A22:      t in A and
A23:      between s,t,r;
A24:      not t in A9
          proof
            assume t in A9;
            then Collinear p,r,s by A23,A1,A22,GTARSKI3:89;
            then s in Line(p,r);
            hence contradiction by A1,A3,A4,A19,GTARSKI3:87;
          end;
          A c= Plane(A9,A) by A1,Th40;
          then
A25:      Plane(A9,t) = Plane(A9,u) by A17,A22,A1,A24,A16,Th34;
A26:      Plane(A9,t) = {x where x is POINT of S : A9 out x,t or
            x in A9 or between t,A9,x} by Th32,A1,A24;
          A9 is_line & Collinear s,t,r & r out s,t
            by A1,A4,A19,A3,A22,A23,GTARSKI3:14;
          then A9 out s,t by A19,A4,Th29;
          hence x in Plane(A9,A) by A26,A17,A25,A14;
        end;
        suppose between r9,A,s;
          then consider t be POINT of S such that
A27:      t in A and
A28:      between r9,t,s;
          r9 in A9 & p in A9 & r9 <> p by A11,A6,A9,A1;
          then
A29:      Line(r9,p) = A9 by A1,GTARSKI3:87;
A30:      not t in A9
          proof
            assume t in A9;
            then Collinear r9,p,s by A27,A28,A1,GTARSKI3:89;
            hence contradiction by A19,A29;
          end;
          A is_line & A9 is_line & A <> A9 & A /\ A9 is non empty
            by A1,XBOOLE_0:def 4;
          then consider t9 be POINT of S such that
A31:      not t9 in A9 and
          t9 in A and
A33:      Plane(A9,A) = Plane(A9,t9) by Def13;
            A c= Plane(A9,A) by A1,Th40;
          then
A34:      Plane(A9,t9) = Plane(A9,t) by A33,A27,A1,A31,A30,Th34;
:::A35:      Plane(A9,t) = Plane(A9,A) by a34,A27,A1,A31,A30,Th34,A33;
          set B = Line(s,r9);
            A9 is_line & B is_line & A9 <> B & r9 in A9 & r9 in B
              by A1,A19,A12,A6,GTARSKI3:83;
            then T1:A9,B Is r9;
            Collinear s,r9,t by A28;
            then t in B;
          then B c= Plane(A9,t) by A9,A27,Th37,T1;
          hence x in Plane(A9,A) by A14,A34,A33,GTARSKI3:83;
        end;
      end;
    end;
    hence thesis;
  end;
