let GX be non empty TopSpace; :: thesis: for B being Subset of GX
for V being Subset of (GX | B) holds Cl V = (Cl (Up V)) /\ B
let B be Subset of GX; :: thesis: for V being Subset of (GX | B) holds Cl V = (Cl (Up V)) /\ B
let V be Subset of (GX | B); :: thesis: Cl V = (Cl (Up V)) /\ B
A1: B = the carrier of (GX | B)

proof

B = [#] (GX | B) by PRE_TOPC:def 10;
hence B = the carrier of (GX | B) ; :: thesis: verum

end;

then Cl (Down (Up V),B) = (Cl (Up V)) /\ B by Th30;
hence Cl V = (Cl (Up V)) /\ B by A1, XBOOLE_1:28; :: thesis: verum