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 Th64:
  Collinear a,b,x & Collinear c,d,x & a <> x & c <> x implies
  a,b,c,d are_coplanar
  proof
    assume that
A1: Collinear a,b,x and
A2: Collinear c,d,x and
A3: a <> x and
A4: c <> x;
    per cases;
    suppose
A5:   not Collinear a,c,x;
      then consider E be Subset of S such that
A6:   E = Plane(a,c,x) and
A7:   E is_plane and
A8:   a in E and
A9:   c in E and x in E by Th49;
T1: not c in Line(a,x)
        proof
          assume c in Line(a,x);
          then ex y be POINT of S st c = y & Collinear a,x,y;
          hence contradiction by GTARSKI3:14,A5;
        end;
        Line(a,x) is_line by A3;
      then
A10:  Line(a,x) c= Plane(Line(a,x),c) by T1,Th31;
      E = Plane(a,x,c) & not Collinear a,x,c by A6,Th53,A5,GTARSKI3:14;
      then
A11: E = Plane(Line(a,x),c) by Def11;
      E = Plane(c,x,a) & not Collinear c,x,a by A5,A6,Th53;
      then
A12:  E = Plane(Line(c,x),a) by Def11;
       Y1: not a in Line(c,x)
        proof
          assume a in Line(c,x);
          then ex y be POINT of S st a = y & Collinear c,x,y;
          hence contradiction by A5;
        end;
         Line(c,x) is_line by A4;
      then
A13:  Line(c,x) c= E by Y1,A12,Th31;
        Collinear a,x,b by A1,GTARSKI3:14;
        then T1: b in Line(a,x);
        Collinear c,x,d by A2,GTARSKI3:14;
        then d in Line(c,x);
      hence thesis by T1,A13,A10,A11,A7,A8,A9;
    end;
    suppose
A14:  Collinear a,c,x;
      set A = Line(a,x);
        Y1:A is_line by A3;
        Collinear a,x,c by A14,GTARSKI3:14;
        then Y3: c in Line(a,x);
        x in A by GTARSKI3:83;
      then
A15:  Line(a,x) = Line(c,x) by Y1,A4,Y3,GTARSKI3:87;
A16:  Collinear a,x,b by A1,GTARSKI3:14;
      Collinear c,x,d by A2,GTARSKI3:14;
      then
A17:  a in A & b in A & c in A & d in A by A16,A15,GTARSKI3:83;
      consider p be POINT of S such that
A18:  not Collinear a,x,p by A3,GTARSKI3:92;
        T1: not p in A
        proof
          assume p in A;
          then ex y be POINT of S st p = y  & Collinear a,x,y;
          hence contradiction by A18;
        end;
        A is_line by A3;
      then
A19:  A c= Plane(A,p) by T1,Th31;
      set E = Plane(Line(a,x),p);
A20:  E = Plane(a,x,p) by A18,Def11;
      now
        take E;
        thus E is_plane by A20,A18;
        thus a in E & b in E & c in E & d in E by A17,A19;
      end;
      hence thesis;
    end;
  end;
