theorem Th55:
  union BOOL X = X
proof
  {} c= X;
  then BOOL X = bool X \ {{}} & {{}} c= bool X by ORDERS_1:def 3,ZFMISC_1:31;
  then
A1: BOOL X \/ {{}} c= bool X by XBOOLE_1:8;
  BOOL X \/ {{}} = (bool X \{{}}) \/{{}} by ORDERS_1:def 3
    .= bool X \/{{}} by XBOOLE_1:39;
  then bool X c= BOOL X \/ {{}} by XBOOLE_1:7;
  then
A2: bool X = BOOL X \/ {{}} by A1,XBOOLE_0:def 10;
  X = union bool X by ZFMISC_1:81
    .= union BOOL X \/ union {{}} by A2,ZFMISC_1:78
    .= union BOOL X \/ {} by ZFMISC_1:25;
  hence thesis;
end;
