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

theorem Th75:
  lim_inf (A (\/) A1) = A \/ lim_inf A1
proof
  reconsider X1 = (inferior_setsequence(A1)) as SetSequence of X;
  reconsider X2 = (inferior_setsequence(A (\/) A1)) as SetSequence of X;
  X2 = A (\/) X1
  proof
    let n be Element of NAT;
    thus X2.n = A \/ X1.n by Th51
      .= (A (\/) X1).n by Def6;
  end;
  then Union X2 = A \/ Union X1 by Th39;
  then lim_inf (A (\/) A1) = A \/ Union X1 by SETLIM_1:def 4;
  hence thesis by SETLIM_1:def 4;
end;
