reserve X for TopStruct,
  A for Subset of X;

theorem Th3:
  for X being TopSpace, A,B being Subset of X holds Int A /\ Cl B c= Cl(A /\ B)
proof
  let X be TopSpace, A,B be Subset of X;
  (Int A) /\ B c= A /\ B by TOPS_1:16,XBOOLE_1:26;
  then
A1: Cl((Int A) /\ B) c= Cl(A /\ B) by PRE_TOPC:19;
  Int A /\ Cl B c= Cl((Int A) /\ B) by TOPS_1:13;
  hence thesis by A1,XBOOLE_1:1;
end;
