reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;

theorem Th2:
  for F,G st F c= G & G is discrete holds F is discrete
proof
  let F,G be Subset-Family of T;
  assume that
A1: F c= G and
A2: G is discrete;
  now
    let p;
    thus ex O st p in O & for C,D st C in F & D in F holds O meets C & O
    meets D implies C=D
    proof
      consider U such that
A3:   p in U & for C,D st C in G & D in G holds U meets C & U meets D
      implies C=D by A2;
      take O=U;
      thus thesis by A1,A3;
    end;
    hence
    ex O st p in O & for C,D st C in F & D in F holds O meets C & O meets
    D implies C=D;
  end;
  hence thesis;
end;
