take X = {{}}; :: thesis: X is cap-finite-closed
now :: thesis: for SX being non empty finite Subset of X holds meet SX in X
let SX be non empty finite Subset of X; :: thesis: meet SX in X
( meet SX = meet {{}} or meet SX = meet {} ) by ZFMISC_1:33;
then meet SX = {} by SETFAM_1:10, SETFAM_1:1;
hence meet SX in X by TARSKI:def 1; :: thesis: verum
end;
hence X is cap-finite-closed ; :: thesis: verum