
theorem Th18:
  for X being set, A being Subset-Family of X st A = {{},X}
  holds UniCl A = A & FinMeetCl A = A
proof
  let X be set, A be Subset-Family of X such that
A1: A = {{},X};
  hereby
    let a be object;
    assume a in UniCl A;
    then consider y being Subset-Family of X such that
A2: y c= A and
A3: a = union y by CANTOR_1:def 1;
    y = {} or y = {{}} or y = {X} or y = {{},X} by A1,A2,ZFMISC_1:36;
    then a = {} or a = X or a = {} \/ X & {} \/ X = X by A3,ZFMISC_1:2,25,75;
    hence a in A by A1,TARSKI:def 2;
  end;
  thus A c= UniCl A by CANTOR_1:1;
  hereby
    let a be object;
    assume a in FinMeetCl A;
    then consider y being Subset-Family of X such that
A4: y c= A and y is finite and
A5: a = Intersect y by CANTOR_1:def 3;
    y = {} or y = {{}} or y = {X} or y = {{},X} by A1,A4,ZFMISC_1:36;
    then a = X or a = meet {{}} or a = meet {X} or a = meet {{},X}
    by A5,SETFAM_1:def 9;
    then a = X or a = {} or a = {} /\ X & {} /\ X = {} by SETFAM_1:10,11;
    hence a in A by A1,TARSKI:def 2;
  end;
  thus thesis by CANTOR_1:4;
end;
