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

theorem Th82:
  lim_sup (A (\/) A1) = A \/ lim_sup A1
proof
  reconsider X1 = (superior_setsequence(A1)) as SetSequence of X;
  reconsider X2 = (superior_setsequence(A (\/) A1)) as SetSequence of X;
  X2 = A (\/) X1
  proof
    let n be Element of NAT;
    thus X2.n = A \/ X1.n by Th56
      .= (A (\/) X1).n by Def6;
  end;
  then Intersection X2 = A \/ Intersection X1 by Th34;
  then lim_sup (A (\/) A1) = A \/ Intersection X1 by SETLIM_1:def 5;
  hence thesis by SETLIM_1:def 5;
end;
