
theorem
  for X being set, A, B, C being SetSequence of X
    st for n being Nat holds C.n = A.n \/ B.n
   holds lim_sup C = lim_sup A \/ lim_sup B
proof
  let X be set, A, B, C be SetSequence of X;
  assume
A1: for n being Nat holds C.n = A.n \/ B.n;
  thus lim_sup C c= lim_sup A \/ lim_sup B
  proof
    let x be object;
    assume
A2: x in lim_sup C;
    (for n being Nat ex k being Nat st x in A.(n+k))
    or for n being Nat ex k being Nat st x in B.(n+k)
    proof
      given n1 being Nat such that
A3:   for k being Nat holds not x in A.(n1+k);
      given n2 being Nat such that
A4:   for k being Nat holds not x in B.(n2+k);
      set n = max (n1, n2);
      consider g being Nat such that
A5:   n = n1 + g by NAT_1:10,XXREAL_0:25;
      consider h being Nat such that
A6:   n = n2 + h by NAT_1:10,XXREAL_0:25;
      reconsider n as Nat by TARSKI:1;
      consider k being Nat such that
A7:   x in C.(n+k) by A2,Th5;
A8:   x in A.(n+k) \/ B.(n+k) by A1,A7;
      per cases by A8,XBOOLE_0:def 3;
      suppose
        x in A.(n+k);
        then x in A.(n1+(g+k)) by A5;
        hence thesis by A3;
      end;
      suppose
        x in B.(n+k);
        then x in B.(n2+(h+k)) by A6;
        hence thesis by A4;
      end;
    end;
    then x in lim_sup A or x in lim_sup B by Th5;
    hence thesis by XBOOLE_0:def 3;
  end;
  thus lim_sup A \/ lim_sup B c= lim_sup C
  proof
    let x be object;
    assume
A9: x in lim_sup A \/ lim_sup B;
    per cases by A9,XBOOLE_0:def 3;
    suppose
A10:  x in lim_sup A;
      for n being Nat ex k being Nat st x in C.(n+k )
      proof
        let n be Nat;
        consider k being Nat such that
A11:    x in A.(n+k) by A10,Th5;
        take k;
        x in A.(n+k) \/ B.(n+k) by A11,XBOOLE_0:def 3;
        hence thesis by A1;
      end;
      hence thesis by Th5;
    end;
    suppose
A12:  x in lim_sup B;
      for n being Nat ex k being Nat st x in C.(n+k )
      proof
        let n be Nat;
        consider k being Nat such that
A13:    x in B.(n+k) by A12,Th5;
        take k;
        x in A.(n+k) \/ B.(n+k) by A13,XBOOLE_0:def 3;
        hence thesis by A1;
      end;
      hence thesis by Th5;
    end;
  end;
end;
