let GX be non empty TopSpace; :: thesis: for B1, B2, V being Subset of GX holds Down (B1 /\ B2),V = (Down B1,V) /\ (Down B2,V)
let B1, B2, V be Subset of GX; :: thesis: Down (B1 /\ B2),V = (Down B1,V) /\ (Down B2,V)
A1: ( Down (B1 /\ B2),V = (B1 /\ B2) /\ V & Down B1,V = B1 /\ V & Down B2,V = B2 /\ V ) by CONNSP_3:def 5;
then Down (B1 /\ B2),V = B1 /\ (B2 /\ (V /\ V)) by XBOOLE_1:16
.= B1 /\ ((B2 /\ V) /\ V) by XBOOLE_1:16
.= (B1 /\ V) /\ (B2 /\ V) by XBOOLE_1:16 ;
hence Down (B1 /\ B2),V = (Down B1,V) /\ (Down B2,V) by A1; :: thesis: verum