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 Th3:
  Line(Pn,Pn)={Pn}
  proof
    now
      hereby
        let x be object;
        assume x in Line(Pn,Pn);
        then consider lambda be Real such that
A1:     x = (1-lambda) * Pn + lambda * Pn;
        x = ((1 - lambda)+ lambda) * Pn by A1,EUCLID_4:7
         .= Pn by EUCLID_4:3;
        hence x in {Pn} by TARSKI:def 1;
      end;
      let x be object;
      assume x in {Pn};
      then x = Pn by TARSKI:def 1;
      hence x in Line(Pn,Pn) by EUCLID_4:9;
    end;
    then Line(Pn,Pn) c= {Pn} & {Pn} c= Line(Pn,Pn);
    hence thesis;
  end;
