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

theorem Th64:
  A2 is convergent implies lim_inf (A1 (\) A2) = lim_inf A1 \ lim_inf A2
proof
  assume
A1: A2 is convergent;
  thus lim_inf (A1 (\) A2) c= lim_inf A1 \ lim_inf A2 by Th62;
  thus lim_inf A1 \ lim_inf A2 c= lim_inf (A1 (\) A2)
  proof
    let x be object;
    assume
A2: x in lim_inf A1 \ lim_inf A2;
    then x in lim_inf A1 by XBOOLE_0:def 5;
    then consider n0 being Nat such that
A3: for k holds x in A1.(n0+k) by KURATO_0:4;
    not x in lim_inf A2 by A2,XBOOLE_0:def 5;
    then not x in lim_sup A2 by A1,KURATO_0:def 5;
    then consider n1 being Nat such that
A4: for k holds not x in A2.(n1+k) by KURATO_0:5;
    now
      let k;
      x in A1.(n0+(n1+k)) & not x in A2.(n1+(n0+k)) by A3,A4;
      then x in A1.((n0+n1)+k) \ A2.((n0+n1)+k) by XBOOLE_0:def 5;
      hence x in (A1 (\) A2).((n0+n1)+k) by Def3;
    end;
    hence thesis by KURATO_0:4;
  end;
end;
