theorem Th2:
  X is closed_wrt_A1-A7 implies (o in X iff {o} in X) & (A in X
  implies union A in X)
proof
  assume
A1: X is closed_wrt_A1-A7;
  then
A2: X is closed_wrt_A2;
A3: now
    assume o in X;
    then {o,o} in X by A2;
    hence {o} in X by ENUMSET1:29;
  end;
A4: X is closed_wrt_A3 by A1;
  now
    assume {o} in X;
    then union {o} in X by A4;
    hence o in X by ZFMISC_1:25;
  end;
  hence o in X iff {o} in X by A3;
  assume A in X;
  hence thesis by A4;
end;
