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 Th53:
  not Collinear a,b,c implies
  Plane(a,b,c) = Plane(b,c,a) & Plane(a,b,c) = Plane(c,a,b) &
  Plane(a,b,c) = Plane(b,a,c) & Plane(a,b,c) = Plane(a,c,b) &
  Plane(a,b,c) = Plane(c,b,a)
  proof
    assume
A1: not Collinear a,b,c;
    then
A2: not Collinear b,c,a & not Collinear c,a,b & not Collinear b,a,c &
    not Collinear a,c,b & not Collinear c,b,a by GTARSKI3:45;
    consider Eabc be Subset of S such that
A3: Plane(a,b,c) = Eabc & Eabc is_plane & a in Eabc & b in Eabc & c in Eabc
      by A1,Th49;
    consider Ebca be Subset of S such that
A4: Plane(b,c,a) = Ebca & Ebca is_plane & b in Ebca & c in Ebca & a in Ebca
      by A2,Th49;
    consider Ecab be Subset of S such that
A5: Plane(c,a,b) = Ecab & Ecab is_plane & c in Ecab & a in Ecab & b in Ecab
      by A2,Th49;
    consider Ebac be Subset of S such that
A6: Plane(b,a,c) = Ebac & Ebac is_plane & b in Ebac & a in Ebac & c in Ebac
      by A2,Th49;
    consider Eacb be Subset of S such that
A7: Plane(a,c,b) = Eacb & Eacb is_plane & a in Eacb & c in Eacb & b in Eacb
      by A2,Th49;
    consider Ecba be Subset of S such that
A8: Plane(c,b,a) = Ecba & Ecba is_plane & c in Ecba & b in Ecba & a in Ecba
      by A2,Th49;
    thus thesis by A1,A3,A4,A5,A6,A7,A8,Th52;
  end;
