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 Th2:
  Euclid n = MetricSpaceNorm REAL-NS n
  proof
    set X = REAL-NS n;
    A1: the carrier of Euclid n
      = the carrier of X by REAL_NS1:def 4;

    for x,y be Element of REAL n holds
        (the distance of Euclid n).(x,y)
      = (distance_by_norm_of X).(x,y)
    proof
      let x,y be Element of REAL n;
      reconsider x1=x, y1=y as Point of REAL-NS n by REAL_NS1:def 4;
      thus (the distance of Euclid n).(x,y)
       = |. x-y .| by EUCLID:def 6
      .= ||. x1-y1 .|| by REAL_NS1:1,5
      .= (distance_by_norm_of X).(x,y) by NORMSP_2:def 1;
    end;
    hence thesis by BINOP_1:2,A1;
  end;
