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,G,H st F is discrete & G is discrete & INTERSECTION(F,G)=H holds
  H is discrete
proof
  let F,G,H;
  assume that
A1: F is discrete and
A2: G is discrete and
A3: INTERSECTION(F,G)=H;
  now
    let p;
    consider O such that
A4: p in O and
A5: for A,B being Subset of T st A in F & B in F holds O meets A & O
    meets B implies A=B by A1;
    consider U such that
A6: p in U and
A7: for A,B being Subset of T st A in G & B in G holds U meets A & U
    meets B implies A=B by A2;
A8: now
      let A,B;
      assume that
A9:   A in INTERSECTION(F,G) and
A10:  B in INTERSECTION(F,G);
      consider af,ag being set such that
A11:  af in F and
A12:  ag in G and
A13:  A=af/\ag by A9,SETFAM_1:def 5;
      assume that
A14:  (O/\U) meets A and
A15:  (O/\U) meets B;
      consider bf,bg being set such that
A16:  bf in F and
A17:  bg in G and
A18:  B=bf/\bg by A10,SETFAM_1:def 5;
      (O/\U)/\(bf/\bg)<>{} by A18,A15,XBOOLE_0:def 7;
      then (O/\U/\bf) /\bg <>{} by XBOOLE_1:16;
      then
A19:  (O/\bf/\U)/\bg<>{} by XBOOLE_1:16;
      then O/\bf <>{};
      then
A20:  O meets bf by XBOOLE_0:def 7;
      (O/\U)/\(af/\ag)<>{} by A13,A14,XBOOLE_0:def 7;
      then (O/\U/\af)/\ag<>{} by XBOOLE_1:16;
      then
A21:  (O/\af/\U)/\ag <> {} by XBOOLE_1:16;
      then (O/\af)/\(U/\ag) <> {} by XBOOLE_1:16;
      then U/\ag <> {};
      then
A22:  U meets ag by XBOOLE_0:def 7;
      (O/\bf)/\(U/\bg)<>{} by A19,XBOOLE_1:16;
      then U/\bg <>{};
      then
A23:  U meets bg by XBOOLE_0:def 7;
      O/\af <>{} by A21;
      then O meets af by XBOOLE_0:def 7;
      then af=bf by A5,A11,A16,A20;
      hence A=B by A7,A12,A13,A17,A18,A22,A23;
    end;
    set Q=O/\U;
    p in Q by A4,A6,XBOOLE_0:def 4;
    hence ex Q being open Subset of T st p in Q & for A,B being Subset of T st
    A in H & B in H holds Q meets A & Q meets B implies A=B by A3,A8;
  end;
  hence thesis;
end;
