reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

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;
