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 Th4:
  An=PAn & Bn=PBn implies Line(An,Bn)=Line(PAn,PBn)
  proof
    assume
A1: An=PAn & Bn=PBn;
    now
      hereby
        let x be object;
        assume x in Line(An,Bn);
        then x in the set of all (1-lambda)*An + lambda*Bn
                  where lambda is Real by RLTOPSP1:def 14;
        then consider l0 be Real such that
A2:     x = (1-l0)*An+l0*Bn;
        thus x in Line(PAn,PBn) by A1,A2;
      end;
      let x be object;
      assume x in Line(PAn,PBn);
      then consider l0 be Real such that
A3:   x = (1-l0)*PAn+l0*PBn;
      x = (1-l0)*An+l0*Bn by A1,A3;
      then x in the set of all (1-lambda)*An + lambda*Bn where lambda is Real;
      hence x in Line(An,Bn) by RLTOPSP1:def 14;
    end;
    then Line(An,Bn) c= Line(PAn,PBn) & Line(PAn,PBn) c= Line(An,Bn);
    hence thesis;
  end;
