reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th6:
  for Nrm be RealNormSpace,
        X be Subset of Nrm
    st X is compact
  holds
    X is closed
  & ex r be Real
    st for y be Point of Nrm
        st y in X
       holds ||.y.|| < r
  proof
    let Nrm be RealNormSpace,
          X be Subset of Nrm;
    assume
    A1: X is compact;

    per cases;
    suppose
      A2: X = {};
      reconsider Y = X as Subset of TopSpaceNorm Nrm;
      Y is closed by A2;
      hence X is closed by NORMSP_2:15;
      thus
      ex r be Real
      st for y be Point of Nrm
          st y in X
         holds ||.y.|| < r
      proof
        take 1;
        thus for y be Point of Nrm st y in X holds ||.y.|| < 1 by A2;
      end;
    end;

    suppose
      A3: X <> {};
      thus X is closed
      proof
        assume not X is closed;
        then consider s1 be sequence of Nrm such that
        A4: rng s1 c= X and
        A5: s1 is convergent & not lim s1 in X;
        ex s2 be sequence of Nrm
        st s2 is subsequence of s1
         & s2 is convergent
         & lim s2 in X by A1,A4;
        hence contradiction by A5,LOPBAN_3:8;
      end;

      reconsider Y = X as Subset of MetricSpaceNorm(Nrm);
      A6: Y is sequentially_compact by A1,TOPMETR4:18;
      reconsider Z = Y as Subset of TopSpaceMetr(MetricSpaceNorm(Nrm));

      Z is compact by TOPMETR4:15,A6;
      then
      consider r0 be Real such that
      A7: 0 < r0
        & for x, y be Point of MetricSpaceNorm(Nrm)
           st x in Y & y in Y
          holds dist (x,y) <= r0
            by HAUSDORF:18,TBSP_1:def 7;

      consider x0 be object such that
      A8: x0 in Y by A3,XBOOLE_0:def 1;
      reconsider x0 as Point of MetricSpaceNorm(Nrm) by A8;
      reconsider x = x0 as Point of Nrm;

      set r = 1 + r0 + ||.x.||;
      reconsider r as Real;
      take r;

      thus
      for y be Point of Nrm
       st y in X
      holds ||.y.|| < r
      proof
        let y be Point of Nrm;
        assume
        A9: y in X;
        reconsider y0 = y as Point of MetricSpaceNorm(Nrm);

        y = x+(y-x) by RLVECT_4:1;
        then
        A10: ||.y.|| <= ||.x.|| + ||.y-x.|| by NORMSP_1:def 1;
        ||.y-x.|| = dist (y0,x0) by NORMSP_2:def 1;
        then ||.y-x.|| + 0 < r0 + 1 by A7,A8,A9,XREAL_1:8;
        then ||.x.|| + ||.y-x.|| < ||.x.|| + (r0+1) by XREAL_1:8;
        hence ||.y.|| < r by A10,XXREAL_0:2;
      end;
    end;
  end;
