reserve i,j,k for Nat,
  r,s,r1,r2,s1,s2,sb,tb for Real,
  x for set,
  GX for non empty TopSpace;
reserve GZ for non empty TopSpace;

theorem
  for B1,B2,V being Subset of GX holds Down(B1 /\ B2,V)=Down(B1,V) /\
  Down(B2,V)
proof
  let B1,B2,V be Subset of GX;
  Down(B1 /\ B2,V)=(B1 /\ B2)/\ V by CONNSP_3:def 5;
  then
A1: 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;
  Down(B1,V)=B1 /\ V by CONNSP_3:def 5;
  hence thesis by A1,CONNSP_3:def 5;
end;
