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

theorem Th83:
  A \ lim_sup A1 c= lim_sup (A (\) A1)
proof
  let x be object;
  assume
A1: x in A \ lim_sup A1;
  then not x in lim_sup A1 by XBOOLE_0:def 5;
  then consider n1 being Nat such that
A2: for k holds not x in A1.(n1+k) by KURATO_0:5;
  assume not x in lim_sup (A (\) A1);
  then consider n such that
A3: for k holds not x in (A (\) A1).(n+k) by KURATO_0:5;
A4: now
    let k;
    not x in (A (\) A1).(n+k) by A3;
    then not x in A \ A1.(n+k) by Def7;
    hence not x in A or x in A1.(n+k) by XBOOLE_0:def 5;
  end;
  per cases by A4;
  suppose
    not x in A;
    hence contradiction by A1,XBOOLE_0:def 5;
  end;
  suppose
A5: x in A1.(n+k);
    not x in A1.(n1+n) by A2;
    hence contradiction by A5;
  end;
end;
