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
  for V be finite-dimensional RealNormSpace,
      X be Subset of V
    st dim V <> 0
  holds
    X is compact
      iff
    X is closed
      &
     ex r be Real
     st for y be Point of V st y in X
        holds ||.y.|| < r
  proof
    let V be finite-dimensional RealNormSpace;
    let X be Subset of V;
    assume
    A1: dim V <> 0;

    thus X is compact implies X is closed
      &
    ex r be Real
    st for y be Point of V st y in X
       holds ||.y.|| < r by Th6;

    assume
    A2: X is closed
          &
        ex r be Real
        st for x be Point of V st x in X
           holds ||.x.|| < r;

    then consider r be Real such that
    A3: for x be Point of V st x in X
        holds ||.x.|| < r;

    consider S be LinearOperator of V,REAL-NS(dim V) such that
    A4: S is one-to-one onto isometric-like by A1,Th28;
    consider k1,k2 be Real such that
    A5: 0 <= k1
      & 0 <= k2
      & for x be Element of V
        holds ||.S.x.|| <= k1 * ||.x.||
      & ||.x.|| <= k2*||.S.x.|| by A4;

    A6: S.:X is closed by A2,A4,Th36;
    reconsider r2 = k1 * r + 1 as Real;

    for y be Point of REAL-NS(dim V) st y in S.:X
    holds ||.y.|| < r2
    proof
      let y be Point of REAL-NS(dim V);
      assume y in S.:X;

      then consider x be object such that
      A7: x in dom S
        & x in X
        & y = S.x by FUNCT_1:def 6;

      reconsider x as Point of V by A7;
      A8: ||.S.x.|| <= k1 * ||.x.|| by A5;
      ||.x.|| < r by A3,A7;
      then k1 * ||.x.|| <= k1 * r by XREAL_1:64,A5;
      then ||.S.x.|| <= k1 * r by XXREAL_0:2,A8;
      then ||.S.x.|| + 0 < k1 * r + 1 by XREAL_1:8;
      hence ||.y.|| < r2 by A7;
    end; then
    S.:X is compact by A6,Lm3;
    hence X is compact by A4,Th38;
  end;
