
theorem
  for X being set, A,B being Subset-Family of X st A = B \/ {X} or B = A \ { X
  } holds FinMeetCl A = FinMeetCl B
proof
  let X be set, A,B be Subset-Family of X;
  assume
A1: A = B \/ {X} or B = A \ {X};
  X in FinMeetCl B by CANTOR_1:8;
  then
A2: {X} c= FinMeetCl B by ZFMISC_1:31;
A3: B c= FinMeetCl B by CANTOR_1:4;
  (A \ {X}) \/ {X} = A \/ {X} by XBOOLE_1:39;
  then
A4: A c= B \/ {X} by A1,XBOOLE_1:7;
  B \/ {X} c= FinMeetCl B by A2,A3,XBOOLE_1:8;
  then A c= FinMeetCl B by A4;
  then FinMeetCl A c= FinMeetCl FinMeetCl B by CANTOR_1:14;
  hence FinMeetCl A c= FinMeetCl B by CANTOR_1:11;
  thus thesis by A1,CANTOR_1:14,XBOOLE_1:7,36;
end;
