reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem
  A1 is convergent implies (A1 (\) A) is convergent & lim (A1 (\) A) =
  lim A1 \ A
proof
  assume
A1: A1 is convergent;
A2: lim_inf (A1 (\) A) = lim_inf A1 \ A by Th77
    .= lim A1 \ A by A1,KURATO_0:def 5;
  then lim_sup (A1 (\) A) = lim_inf (A1 (\) A) by Th84;
  hence thesis by A2,KURATO_0:def 5;
end;
