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
  not Collinear a,b,c implies (for E being Subset of S st a in E & b in E &
  c in E & (for u,v being POINT of S st u in E & v in E & u <> v holds
  Line(u,v) c= E) holds Plane(a,b,c) c= E)
  proof
    assume that
A1: not Collinear a,b,c;
    set P = Plane(a,b,c);
A2: Plane(a,b,c) is_plane & a in Plane(a,b,c) & b in Plane(a,b,c) &
      c in Plane(a,b,c) & (for u,v being POINT of S st
      u in Plane(a,b,c) & v in Plane(a,b,c) & u <> v holds
      Line(u,v) c= Plane(a,b,c)) by A1,Th69;
    let E be Subset of S;
    assume that
A3: a in E and
A4: b in E and
A5: c in E and
A6: (for u,v being POINT of S st u in E & v in E & u <> v holds
      Line(u,v) c= E);
A7: a <> b by A1,GTARSKI3:46;
A8: a <> c
    proof
      assume a = c;
      then Collinear a,c,b by GTARSKI3:46;
      hence contradiction by A1,GTARSKI3:14;
    end;
A9: b <> c
    proof
      assume b = c;
      then Collinear b,c,a by GTARSKI3:46;
      hence contradiction by A1;
    end;
    Plane(a,b,c) c= E
    proof
      let x be object;
      assume
A10:  x in Plane(a,b,c);
      then reconsider d = x as POINT of S;
      a,b,c,d are_coplanar by A10,A2;
      then consider y be POINT of S such that
A11:  (Collinear a,b,y & Collinear c,d,y) or
        (Collinear a,c,y & Collinear b,d,y) or
        (Collinear a,d,y & Collinear b,c,y) by Th68;
      per cases by A11;
      suppose
A12:    Collinear a,b,y & Collinear c,d,y;
        then
A13:    y in Line(a,b);
A14:    Line(a,b) c= E by A7,A3,A4,A6;
        Collinear c,y,d by A12,GTARSKI3:14;
        then
A15:    d in Line(c,y);
        Line(c,y) c= E by A12,A1,A14,A13,A5,A6;
        hence thesis by A15;
      end;
      suppose
A16:    Collinear a,c,y & Collinear b,d,y;
        then
A17:    y in Line(a,c);
A18:    Line(a,c) c= E by A8,A3,A5,A6;
        Collinear b,y,d by A16,GTARSKI3:14;
        then
A19:    d in Line(b,y);
        b <> y by A16,A1,GTARSKI3:14;
        then Line(b,y) c= E by A18,A17,A4,A6;
        hence thesis by A19;
      end;
      suppose
A20:    Collinear a,d,y & Collinear b,c,y;
        then
A21:    y in Line(b,c);
A22:    Line(b,c) c= E by A9,A4,A5,A6;
        Collinear a,y,d by A20,GTARSKI3:14;
        then
A23:    d in Line(a,y);
        Line(a,y) c= E by A20,A1,A22,A21,A3,A6;
        hence thesis by A23;
      end;
    end;
    hence thesis;
  end;
