 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem Th14:
  distance_by_norm_of REAL-NS n = Pitag_dist n
  proof
    the carrier of REAL-NS n = REAL n by REAL_NS1:def 4;
    then reconsider d1 = distance_by_norm_of REAL-NS n as
     Function of [:REAL n,REAL n:],REAL;
    now
      let x,y be set;
      assume
A1:   x in REAL n & y in REAL n;
      then x is Element of TOP-REAL n & y is Element of TOP-REAL n
        by EUCLID:22;
      then reconsider px = x,py = y as Element of Euclid n by EUCLID:67;
      reconsider g = x,h = y as Point of REAL-NS n by A1,REAL_NS1:def 4;
      Euclid n = MetrStruct(#REAL n,Pitag_dist n#) by EUCLID:def 7;
      hence (Pitag_dist n).(x,y) = dist(px,py)
                                .= ||.g-h.|| by Th9
                                .= d1.(x,y) by NORMSP_2:def 1;
    end;
    hence thesis by BINOP_1:def 21;
  end;
