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

theorem Th86:
  A1 is convergent implies lim_sup (A (\) A1) = A \ lim_sup A1
proof
  assume
A1: A1 is convergent;
  thus lim_sup (A (\) A1) c= A \ lim_sup A1
  proof
    let x be object;
    assume
A2: x in lim_sup (A (\) A1);
A3: for n ex k st x in A & not x in A1.(n+k)
    proof
      let n;
      consider k such that
A4:   x in (A (\) A1).(n+k) by A2,KURATO_0:5;
      x in A \ A1.(n+k) by A4,Def7;
      then x in A & not x in A1.(n+k) by XBOOLE_0:def 5;
      hence thesis;
    end;
A5: x in A
    proof
A6:   for n holds ex k st x in A
      proof
        let n;
        ex k st x in A & not x in A1.(n+k) by A3;
        hence thesis;
      end;
      assume not x in A;
      then for n holds not x in A;
      hence contradiction by A6;
    end;
    for n holds ex k st not x in A1.(n+k)
    proof
      let n;
      ex k st x in A & not x in A1.(n+k) by A3;
      hence thesis;
    end;
    then not x in lim_inf A1 by KURATO_0:4;
    then not x in lim_sup A1 by A1,KURATO_0:def 5;
    hence thesis by A5,XBOOLE_0:def 5;
  end;
  thus thesis by Th83;
end;
