
theorem Th11:
  for X,Y being set, A being Subset-Family of X st A = {Y}
  holds FinMeetCl A = {Y,X} & UniCl A = {Y,{}}
proof
  let X,Z be set, A be Subset-Family of X such that
A1: A = {Z};
  thus FinMeetCl A c= {Z,X}
  proof
    let x be object;
    assume x in FinMeetCl A;
    then consider Y being Subset-Family of X such that
A2: Y c= A and Y is finite and
A3: x = Intersect Y by CANTOR_1:def 3;
    Y = {} or Y = {Z} by A1,A2,ZFMISC_1:33;
    then x = X or x = meet {Z} by A3,SETFAM_1:def 9;
    then x = X or x = Z by SETFAM_1:10;
    hence thesis by TARSKI:def 2;
  end;
  reconsider E = {} as Subset-Family of X by XBOOLE_1:2;
  reconsider E as Subset-Family of X;
  hereby
    let x be object;
    assume x in {Z,X};
    then x = X or x = Z by TARSKI:def 2;
    then x = X or x = meet {Z} by SETFAM_1:10;
    then x = Intersect E & E c= A or x = Intersect A & A c= A
    by A1,SETFAM_1:def 9;
    hence x in FinMeetCl A by A1,CANTOR_1:def 3;
  end;
  hereby
    let x be object;
    assume x in UniCl A;
    then consider Y being Subset-Family of X such that
A4: Y c= A and
A5: x = union Y by CANTOR_1:def 1;
    Y = {} or Y = {Z} by A1,A4,ZFMISC_1:33;
    then x = {} or x = Z by A5,ZFMISC_1:2,25;
    hence x in {Z,{}} by TARSKI:def 2;
  end;
  let x be object;
  assume x in {Z,{}};
  then x = {} or x = Z by TARSKI:def 2;
  then x = union E & E c= A or x = union A & A c= A by A1,ZFMISC_1:2
,25;
  hence thesis by CANTOR_1:def 1;
end;
