reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem Th32:
  for F, G being set holds meet UNION (F,G) c= meet F \/ meet G
proof
  let F, G be set;
  per cases;
  suppose
A1: F <> {} & G <> {};
    let x be object;
    assume
A2: x in meet UNION (F,G);
    assume
A3: not x in meet F \/ meet G;
    then not x in meet F by XBOOLE_0:def 3;
    then consider Y being set such that
A4: Y in F & not x in Y by A1,SETFAM_1:def 1;
    not x in meet G by A3,XBOOLE_0:def 3;
    then consider Z being set such that
A5: Z in G & not x in Z by A1,SETFAM_1:def 1;
    ( not x in Y \/ Z)& Y \/ Z in UNION (F,G) by A4,A5,SETFAM_1:def 4
,XBOOLE_0:def 3;
    hence thesis by A2,SETFAM_1:def 1;
  end;
  suppose
    F = {} or G = {};
    then UNION (F,G) = {} by Th29;
    then meet UNION (F,G) = {} by SETFAM_1:def 1;
    hence thesis;
  end;
end;
