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
  for A being non empty Subset of TOP-REAL n, p being Point of TOP-REAL
n st for q being Point of TOP-REAL n st q in A holds dist(p,q) >= r holds dist(
  p,A) >= r
proof
  let A be non empty Subset of TOP-REAL n, p9 be Point of TOP-REAL n such that
A1: for q being Point of TOP-REAL n st q in A holds dist(p9,q) >= r;
  for p,q being Point of TOP-REAL n st p in {p9} & q in A holds dist(p,q) >= r
  proof
    let p,q be Point of TOP-REAL n such that
A2: p in {p9} and
A3: q in A;
    p = p9 by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  hence thesis by Th40;
end;
