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;
