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 Th28:
for V be finite-dimensional RealNormSpace
  st dim V <> 0
holds
  ex L be LinearOperator of V,REAL-NS(dim V)
  st L is one-to-one onto isometric-like
proof
  let V be finite-dimensional RealNormSpace;
  assume dim V <> 0; then
  consider k1,k2 be Real,
           S be LinearOperator of V, REAL-NS(dim V) such that
  A1: S is bijective
    & 0 <= k1
    & 0 <= k2
    & for x be Element of V
      holds
        ||.S.x.|| <= k1 * ||.x.||
      & ||.x.|| <= k2 * ||.S.x.|| by Th27;
  take S;
  thus S is one-to-one onto by A1;
  thus S is isometric-like by A1;
end;
