reserve x,y,X for set;

theorem
  for T being non empty TopSpace for F being Subset-Family of T st F is
  closed holds FinMeetCl F is closed
proof
  let T be non empty TopSpace;
  let F be Subset-Family of T such that
A1: F is closed;
  now
    let P be Subset of T;
    assume P in FinMeetCl F;
    then consider Y being Subset-Family of T such that
A2: Y c= F and
    Y is finite and
A3: P = Intersect Y by CANTOR_1:def 3;
A4: P = the carrier of T & the carrier of T = [#]T or P = meet Y by A3,
SETFAM_1:def 9;
    for A being Subset of T st A in Y holds A is closed by A1,A2;
    hence P is closed by A4,PRE_TOPC:14;
  end;
  hence thesis;
end;
