
theorem Th17:
  for X being set, A being empty Subset-Family of X holds FinMeetCl A = {X}
proof
  let X be set, A be empty Subset-Family of X;
  hereby
    let x be object;
    assume x in FinMeetCl A;
    then consider B being Subset-Family of X such that
A1: B c= A and B is finite and
A2: x = Intersect B by CANTOR_1:def 3;
    B = {} by A1;
    then x = X by A2,SETFAM_1:def 9;
    hence x in {X} by TARSKI:def 1;
  end;
  let x be object;
  assume x in {X};
  then
A3: x = X by TARSKI:def 1;
  Intersect {}bool X = X by SETFAM_1:def 9;
  hence thesis by A3,CANTOR_1:def 3;
end;
