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 Th41:
  A,A9 Is p implies ex r being POINT of S st (not r in A & r in A9 &
  Plane(A,A9) = Plane(A,r) & A9 = Line(r,p) & 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)))
  proof
    assume
A1: A,A9 Is p;
    then 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
A2: not r in A and
A3: r in A9 and
A4: Plane(A,A9) = Plane(A,r) by Def13;
A5: Line(r,p) = A9 by A1,A2,A3,GTARSKI3:87;
    consider r9 be POINT of S such that
A6: between r,p,r9 and
A7: p <> r9 by GTARSKI3:36;
A8:    Collinear r,p,r9 by A6;
A9: not r9 in A
    proof
      assume r9 in A;
      then A is_line & A9 is_line & A <> A9 & p in A & r9 in A &
        p in A9 & r9 in A9 by A8,A5,A1;
      hence contradiction by A7,GTARSKI3:89;
    end;
    t3:r9 in A9 & A9 c= Plane(A,r) by A1,A2,A3,A5,A8,Th37;
    Plane(A,r) = Plane(A,r9) by A9,Th34,A1,A2,t3;
    hence thesis by A1,A2,A3,A4,A5,A6,A8,A9;
  end;
