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 Th47:
  not Collinear a,b,c & E is_plane & a in E & b in E & c in E implies
  E = Plane(a,b,c)
  proof
    assume that
A1: not Collinear a,b,c and
A2: E is_plane and
A3: a in E and
A4: b in E and
A5: c in E;
    set A = Line(a,b);
A6: a <> b by A1,GTARSKI3:46;
    then consider r be POINT of S such that
A7: not Collinear a,b,r and
A8: E = Plane(a,b,r) by A2,A3,A4,Th46;
A9: not c in A
    proof
      assume c in A;
      then ex x be POINT of S st c = x & Collinear a,b,x;
      hence contradiction by A1;
    end;
      H1: A is_line by A6;
      H2: not r in A
      proof
        assume r in A;
        then ex x be POINT of S st r = x & Collinear a,b,x;
        hence contradiction by A7;
      end;
       c in Plane(A,r) by A5,A7,A8,Def11;
    then Plane(A,c) = Plane(A,r) by Th34,H1,H2,A9
                   .= E by A7,A8,Def11;
    hence thesis by Def11,A1;
  end;
