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 Th40:
  for A,B being non empty Subset of TOP-REAL n st for p,q being
Point of TOP-REAL n st p in A & q in B holds dist(p,q) >= r holds dist_min(A,B)
  >= r
proof
  let A,B be non empty Subset of TOP-REAL n such that
A1: for p,q being Point of TOP-REAL n st p in A & q in B holds dist(p,q) >= r;
A2: for p,q being Point of Euclid n st p in A & q in B holds dist(p,q) >= r
  proof
    let a,b being Point of Euclid n such that
A3: a in A & b in B;
    reconsider p =a, q = b as Point of TOP-REAL n by TOPREAL3:8;
    ex a, b being Point of Euclid n st p = a & q = b & dist(p,q) = dist(a,
    b) by TOPREAL6:def 1;
    hence thesis by A1,A3;
  end;
  ex A9,B9 be Subset of TopSpaceMetr Euclid n st A = A9 & B = B9 &
  dist_min(A,B) = min_dist_min(A9,B9) by Def1;
  hence thesis by A2,Th13;
end;
