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 Th50:
  for A being non empty Subset of TOP-REAL n, p,q being Point of
  TOP-REAL n st q in A holds dist(p,A) <= dist(p,q)
proof
  let A be non empty Subset of TOP-REAL n;
  let p,q be Point of TOP-REAL n;
  assume q in A;
  then {q} c= A by ZFMISC_1:31;
  then dist(p,A) <= dist(p,{q}) by Th41;
  hence thesis by Th43;
end;
