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

theorem Th9:
  for x,y being Element of Euclid n,
      g,h being Point of REAL-NS n st x = g & y = h holds
  dist(x,y) = ||. g - h .||
  proof
    let x,y be Element of Euclid n, g,h be Point of REAL-NS n;
    assume
A1: x = g & y = h;
    x in Euclid n & y in Euclid n;
    then x in TOP-REAL n & y in TOP-REAL n by EUCLID:67;
    then reconsider rx = x, ry = y as Element of REAL n by EUCLID:22;
A2: Euclid n = MetrStruct(#REAL n,Pitag_dist n#) by EUCLID:def 7;
    reconsider g1 = g, h1 = h as Element of REAL n by REAL_NS1:def 4;
    reconsider z1 = g1 - h1 as Element of REAL n;
    ||.g - h.|| = (the normF of (REAL-NS n)).(g1 - h1) by REAL_NS1:5
               .= (Euclid_norm n).z1 by REAL_NS1:def 4
               .= |.z1.| by REAL_NS1:def 3;
    hence thesis by A1,A2,EUCLID:def 6;
  end;
