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
  for F st F is discrete holds for A,B st A in F & B in F holds Cl(A/\B)
  =Cl A /\ Cl B
proof
  let F such that
A1: F is discrete;
  let A,B such that
A2: A in F & B in F;
  now
    per cases by A1,A2,Th6;
    suppose
      A misses B;
      then
A3:   A/\B ={}T by XBOOLE_0:def 7;
      (Cl A /\ Cl B)={}
      proof
        assume (Cl A /\ Cl B)<>{};
        then consider x being object such that
A4:     x in (Cl A /\ Cl B) by XBOOLE_0:def 1;
        consider O such that
A5:     x in O and
A6:     for A,B st A in F & B in F holds O meets A & O meets B
        implies A=B by A1,A4;
        x in Cl A by A4,XBOOLE_0:def 4;
        then
A7:     O meets A by A5,PRE_TOPC:def 7;
        x in Cl B by A4,XBOOLE_0:def 4;
        then O meets B by A5,PRE_TOPC:def 7;
        then A=B by A2,A6,A7;
        hence thesis by A3,A7,XBOOLE_1:65;
      end;
      hence thesis by A3,PRE_TOPC:22;
    end;
    suppose
      A=B;
      hence thesis;
    end;
  end;
  hence thesis;
end;
