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

theorem Th77:
  lim_inf (A1 (\) A) = lim_inf A1 \ A
proof
  reconsider X1 = (inferior_setsequence(A1)) as SetSequence of X;
  reconsider X2 = (inferior_setsequence(A1 (\) A)) as SetSequence of X;
  X2 = X1 (\) A
  proof
    let n be Element of NAT;
    thus X2.n = X1.n \ A by Th53
      .= (X1 (\) A).n by Def8;
  end;
  then Union X2 = Union X1 \ A by Th41;
  then lim_inf (A1 (\) A) = Union X1 \ A by SETLIM_1:def 4;
  hence thesis by SETLIM_1:def 4;
end;
