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 Th57:
  ex S being ext-real-membered set st S = the set of all |. x.i - y.i .| where
    i is Element of Seg n & (Infty_dist n).(x,y) = sup S
  proof
    set S = the set of all |. x.i - y.i .| where i is Element of Seg n;
A1: S is real-membered & S = rng abs (x-y) by Th56;
    then reconsider S1 = S as ext-real-membered set;
    take S1;
    thus thesis by A1,Def7;
  end;
