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 Th46:
  for A being non empty compact Subset of TOP-REAL n, p being
Point of TOP-REAL n ex q being Point of TOP-REAL n st q in A & dist(p,A) = dist
  (p,q)
proof
  let A be non empty compact Subset of TOP-REAL n;
  let p be Point of TOP-REAL n;
  consider q,p9 being Point of TOP-REAL n such that
A1: q in A and
A2: p9 in {p} & dist_min(A,{p}) = dist(q,p9) by Th42;
  take q;
  thus q in A by A1;
  thus thesis by A2,TARSKI:def 1;
end;
