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

theorem Th84:
  lim_sup (A1 (\) A) = lim_sup A1 \ A
proof
  reconsider X1 = (superior_setsequence(A1)) as SetSequence of X;
  reconsider X2 = (superior_setsequence(A1 (\) A)) as SetSequence of X;
  X2 = X1 (\) A
  proof
    let n be Element of NAT;
    thus X2.n = X1.n \ A by Th58
      .= (X1 (\) A).n by Def8;
  end;
  then Intersection X2 = Intersection X1 \ A by Th36;
  then lim_sup (A1 (\) A) = Intersection X1 \ A by SETLIM_1:def 5;
  hence thesis by SETLIM_1:def 5;
end;
