reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;

theorem Th5:
  An <> Cn & Cn in LSeg(An,Bn) & An in Ln & Cn in Ln & Ln is being_line
  implies Bn in Ln
  proof
    assume that
A1: An <> Cn and
A2: Cn in LSeg(An,Bn) and
A3: An in Ln and
A4: Cn in Ln and
A5: Ln is being_line;
    reconsider rAn=An,rCn=Cn,rBn=Bn as Element of REAL n by EUCLID:22;
    Line(rAn,rCn) = Ln by A1,A3,A4,A5,EUCLIDLP:30;
    then
A6: Line(An,Cn) = Ln by Th4;
    LSeg(An,Bn) c= Line(An,Bn) by RLTOPSP1:73;
    then
A7: Cn in Line(An,Bn) & An in Line(An,Bn) & An <> Cn by A1,A2,EUCLID_4:41;
    Line(rAn,rBn) = Line(An,Bn) & Line(rAn,rCn)=Line(An,Cn) by Th4;
    then Line(An,Bn) c= Line(An,Cn) by A7,EUCLID_4:11;
    hence thesis by A6,EUCLID_4:41;
  end;
