reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;
reserve n for non zero Nat;
reserve n     for non zero Nat,
        x,y,z for Element of REAL n;

theorem Th62:
  Infty_dist n is_metric_of REAL n
  proof
    for x,y,z be Element of REAL n holds
       ((Infty_dist n).(x,y) = 0 iff x = y) &
       ((Infty_dist n).(x,y) = (Infty_dist n).(y,x)) &
       ((Infty_dist n).(x,z) <= (Infty_dist n).(x,y) + (Infty_dist n).(y,z))
      by Th59,Th60,Th61;
    hence thesis by PCOMPS_1:def 6;
  end;
