reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem
  seq1 is bounded & seq2 is bounded implies lim_inf seq1 + lim_inf seq2
  <= lim_inf (seq1+seq2) & lim_inf (seq1+seq2) <= lim_inf seq1 + lim_sup seq2 &
  lim_inf (seq1+seq2) <= lim_sup seq1 + lim_inf seq2 & lim_inf seq1 + lim_sup
seq2 <= lim_sup (seq1+seq2) & lim_sup seq1 + lim_inf seq2 <= lim_sup (seq1+seq2
) & lim_sup (seq1+seq2) <= lim_sup seq1 + lim_sup seq2 & (seq1 is convergent or
seq2 is convergent implies lim_inf (seq1+seq2) = lim_inf seq1 + lim_inf seq2 &
  lim_sup (seq1+seq2) = lim_sup seq1 + lim_sup seq2 )
proof
  assume
A1: seq1 is bounded & seq2 is bounded;
A2: for seq1,seq2 st seq1 is bounded & seq2 is bounded holds lim_sup (seq1+
  seq2) <= lim_sup seq1 + lim_sup seq2
  proof
    let seq1,seq2;
    set r1 = lim_sup seq1, r2 = lim_sup seq2;
    assume that
A3: seq1 is bounded and
A4: seq2 is bounded;
    for s st 0<s holds ex n st for k holds (seq1+seq2).(n+k) < r1+r2+s
    proof
      let s;
      assume 0<s;
      then
A5:   0<s/2 by XREAL_1:215;
      then consider n1 being Nat such that
A6:   for k holds seq1.(n1+k) < r1+s/2 by A3,Th85;
      consider n2 being Nat such that
A7:   for k holds seq2.(n2+k) < r2+s/2 by A4,A5,Th85;
      for k holds (seq1+seq2).(n1+n2+k) < r1+r2+s
      proof
        let k;
        seq1.(n1+(n2+k)) < r1+s/2 & seq2.(n2+(n1+k)) < r2+s/2 by A6,A7;
        then seq1.(n1+n2+k) + seq2.(n1+n2+k) < (r1+s/2)+(r2+s/2) by XREAL_1:8;
        hence thesis by SEQ_1:7;
      end;
      hence thesis;
    end;
    hence thesis by A3,A4,Th85;
  end;
A8: for seq1,seq2 st seq1 is bounded & seq2 is bounded holds lim_inf seq1 +
  lim_sup seq2 <= lim_sup (seq1+seq2)
  proof
    let seq1,seq2;
    assume that
A9: seq1 is bounded and
A10: seq2 is bounded;
    lim_sup ((seq1+seq2)+(-seq1)) <= lim_sup (-seq1) + lim_sup (seq1+
    seq2) by A2,A9,A10;
    then lim_sup (seq1+seq2+-seq1) <= -lim_inf seq1 + lim_sup (seq1+seq2) by A9
,Th90;
    then lim_sup (seq2+seq1-seq1) <= lim_sup (seq1+seq2) - lim_inf seq1;
    then lim_sup seq2 <= lim_sup (seq1+seq2) - lim_inf seq1 by Th2;
    hence thesis by XREAL_1:19;
  end;
  then
A11: lim_inf seq1 + lim_sup seq2 <= lim_sup (seq1+seq2) by A1;
A12: lim_sup seq1 + lim_inf seq2 <= lim_sup (seq1+seq2) by A8,A1;
A13: lim_sup (seq1+seq2) <= lim_sup seq1 + lim_sup seq2 by A2,A1;
A14: for seq1,seq2 st seq1 is bounded & seq2 is bounded holds lim_inf seq1 +
  lim_inf seq2 <= lim_inf (seq1+seq2)
  proof
    let seq1,seq2;
    set r1 = lim_inf seq1, r2 = lim_inf seq2;
    assume that
A15: seq1 is bounded and
A16: seq2 is bounded;
    for s st 0<s holds ex n st for k holds r1+r2-s < (seq1+seq2).(n+k)
    proof
      let s;
      assume 0<s;
      then
A17:  0<s/2 by XREAL_1:215;
      then consider n1 being Nat such that
A18:  for k holds r1-s/2 < seq1.(n1+k) by A15,Th82;
      consider n2 being Nat such that
A19:  for k holds r2-s/2 < seq2.(n2+k) by A16,A17,Th82;
      for k holds r1+r2-s < (seq1+seq2).(n1+n2+k)
      proof
        let k;
        r1-s/2 < seq1.(n1+(n2+k)) & r2-s/2 < seq2.(n2+(n1+k)) by A18,A19;
        then (r1-s/2)+(r2-s/2) < seq1.(n1+n2+k) + seq2.(n1+n2+k) by XREAL_1:8;
        hence thesis by SEQ_1:7;
      end;
      hence thesis;
    end;
    hence thesis by A15,A16,Th82;
  end;
  then
A20: lim_inf seq1 + lim_inf seq2 <= lim_inf (seq1+seq2) by A1;
A21: for seq1,seq2 st seq1 is bounded & seq2 is bounded holds lim_inf (seq1+
  seq2) <= lim_inf seq1 + lim_sup seq2
  proof
    let seq1,seq2;
    assume that
A22: seq1 is bounded and
A23: seq2 is bounded;
    lim_inf ((seq1+seq2)+(-seq2)) >= lim_inf (seq1+seq2) + lim_inf (-
    seq2) by A14,A22,A23;
    then lim_inf (seq1+seq2+-seq2) >= lim_inf (seq1+seq2) +- lim_sup seq2 by
A23,Th90;
    then lim_inf (seq1+seq2-seq2) >= lim_inf (seq1+seq2) - lim_sup seq2;
    then lim_inf seq1 >= lim_inf (seq1+seq2) - lim_sup seq2 by Th2;
    hence thesis by XREAL_1:20;
  end;
  then
A24: lim_inf (seq1+seq2) <= lim_sup seq1 + lim_inf seq2 by A1;
A25: lim_inf (seq1+seq2) <= lim_inf seq1 + lim_sup seq2 by A21,A1;
  seq1 is convergent or seq2 is convergent implies lim_inf (seq1+seq2) =
lim_inf seq1 + lim_inf seq2 & lim_sup (seq1+seq2) = lim_sup seq1 + lim_sup seq2
  proof
    assume
A26: seq1 is convergent or seq2 is convergent;
    per cases by A26;
    suppose
      seq1 is convergent;
      then lim_inf seq1 = lim_sup seq1 by Th88;
      hence thesis by A20,A24,A11,A13,XXREAL_0:1;
    end;
    suppose
      seq2 is convergent;
      then lim_inf seq2 = lim_sup seq2 by Th88;
      hence thesis by A20,A25,A12,A13,XXREAL_0:1;
    end;
  end;
  hence thesis by A14,A21,A2,A8,A1;
end;
