theorem Th43:
  for p,q being Point of TOP-REAL n holds dist_min({p},{q}) = dist (p,q)
proof
  let p,q be Point of TOP-REAL n;
  consider p9,q9 being Point of TOP-REAL n such that
A1: p9 in {p} and
A2: q9 in {q} & dist_min({p},{q}) = dist(p9,q9) by Th42;
  p = p9 by A1,TARSKI:def 1;
  hence thesis by A2,TARSKI:def 1;
end;
