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 & A2 is convergent implies (A1 (\/) A2) is convergent
  & lim (A1 (\/) A2) = lim A1 \/ lim A2
proof
  assume that
A1: A1 is convergent and
A2: A2 is convergent;
A3: lim_sup (A1 (\/) A2) = lim A1 \/ lim A2 by Th68;
  lim_inf (A1 (\/) A2) = lim_inf A1 \/ lim_inf A2 by A1,Th63
    .= lim A1 \/ lim_inf A2 by A1,KURATO_0:def 5
    .= lim A1 \/ lim A2 by A2,KURATO_0:def 5;
  hence thesis by A3,KURATO_0:def 5;
end;
