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

theorem
  lim_sup A1 \+\ lim_sup A2 c= lim_sup (A1 (\+\) A2)
proof
  let x be object;
A1: for A1,A2 st x in lim_sup A1 & not x in lim_sup A2 holds x in lim_sup (
  A1 (\+\) A2)
  proof
    let A1,A2;
    assume that
A2: x in lim_sup A1 and
A3: not x in lim_sup A2;
    consider n1 being Nat such that
A4: for k holds not x in A2.(n1+k) by A3,KURATO_0:5;
    now
      let n;
      consider k1 being Nat such that
A5:   x in A1.((n+n1)+k1) by A2,KURATO_0:5;
      not x in A2.(n1+(n+k1)) by A4;
      then x in A1.(n+(n1+k1)) \+\ A2.(n+(n1+k1)) by A5,XBOOLE_0:1;
      then x in (A1 (\+\) A2).(n+(n1+k1)) by Def4;
      hence ex k st x in (A1 (\+\) A2).(n+k);
    end;
    hence thesis by KURATO_0:5;
  end;
  assume
A6: x in lim_sup A1 \+\ lim_sup A2;
  per cases by A6,XBOOLE_0:1;
  suppose
    x in lim_sup A1 & not x in lim_sup A2;
    hence thesis by A1;
  end;
  suppose
    not x in lim_sup A1 & x in lim_sup A2;
    hence thesis by A1;
  end;
end;
