reserve
  A,B,X for set,
  S for SigmaField of X;

theorem Th3:
  bool X is SigmaField of X
proof
A1: for M being N_Sub_set_fam of X st M c= bool X holds union M in bool X;
  reconsider Y = bool X as Subset-Family of X;
  for A being set holds A in bool X implies X\A in bool X;
  then reconsider
  Y as non empty compl-closed sigma-additive Subset-Family of X by A1,
MEASURE1:def 1,def 5;
  Y is SigmaField of X;
  hence thesis;
end;
