reserve X for non empty set,
  F for with_the_same_dom Functional_Sequence of X, ExtREAL,
  seq,seq1,seq2 for ExtREAL_sequence,
  x for Element of X,
  a,r for R_eal,
  n,m,k for Nat;
reserve S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S;

theorem Th10:
  r in REAL & (for n be Nat holds seq1.n = r + seq2.n) implies
  lim_inf seq1 = r + lim_inf seq2 & lim_sup seq1 = r + lim_sup seq2
proof
  assume that
A1: r in REAL and
A2: for n be Nat holds seq1.n = r + seq2.n;
  reconsider R1 = rng inferior_realsequence seq1, R2 = rng
  inferior_realsequence seq2, P1 = rng superior_realsequence seq1, P2 = rng
  superior_realsequence seq2 as non empty Subset of ExtREAL;
  now
    let z be object;
    assume z in {r} + R2;
    then consider z2,z3 be R_eal such that
A3: z = z2 + z3 and
A4: z2 in {r} and
A5: z3 in R2;
    consider n be object such that
A6: n in NAT and
A7: (inferior_realsequence seq2).n = z3 by A5,FUNCT_2:11;
    reconsider n as Element of NAT by A6;
    consider Y2 be non empty Subset of ExtREAL such that
A8: Y2 = {seq2.k where k is Nat: n <= k} and
A9: z3 = inf Y2 by A7,RINFSUP2:def 6;
    consider Y1 be non empty Subset of ExtREAL such that
A10: Y1 = {seq1.k where k is Nat : n <= k} and
A11: (inferior_realsequence seq1).n = inf Y1 by RINFSUP2:def 6;
    now
      let w be object;
A12:  r in {r} by TARSKI:def 1;
      assume w in Y1;
      then consider k1 be Nat such that
A13:  w = seq1.k1 and
A14:  n <= k1 by A10;
      reconsider w1=w as R_eal by A13;
A15:  seq2.k1 in Y2 by A8,A14;
      w1 = r + seq2.k1 by A2,A13;
      hence w in {r} + Y2 by A12,A15;
    end;
    then
A16: Y1 c= {r} + Y2 by TARSKI:def 3;
    now
      let w be object;
      assume w in {r} + Y2;
      then consider w1,w2 be R_eal such that
A17:  w = w1 + w2 and
A18:  w1 in {r} and
A19:  w2 in Y2;
      consider k2 be Nat such that
A20:  w2 = seq2.k2 and
A21:  n <= k2 by A8,A19;
      w1 = r by A18,TARSKI:def 1;
      then w = seq1.k2 by A2,A17,A20;
      hence w in Y1 by A10,A21;
    end;
    then {r} + Y2 c= Y1 by TARSKI:def 3;
    then Y1 = {r} + Y2 by A16,XBOOLE_0:def 10;
    then inf Y1 = inf{r} + inf Y2 by A1,Th9;
    then inf Y1 = r + inf Y2 by XXREAL_2:13;
    then z = (inferior_realsequence seq1).n by A4,A3,A9,A11,TARSKI:def 1;
    hence z in R1 by FUNCT_2:4;
  end;
  then
A22: {r} + R2 c= R1 by TARSKI:def 3;
  now
    let z be object;
    assume z in {r} + P2;
    then consider z2,z3 be R_eal such that
A23: z = z2 + z3 and
A24: z2 in {r} and
A25: z3 in P2;
    consider n be object such that
A26: n in NAT and
A27: (superior_realsequence seq2).n = z3 by A25,FUNCT_2:11;
    reconsider n as Element of NAT by A26;
    consider Y2 be non empty Subset of ExtREAL such that
A28: Y2 = {seq2.k where k is Nat : n <= k} and
A29: z3 = sup Y2 by A27,RINFSUP2:def 7;
    consider Y1 be non empty Subset of ExtREAL such that
A30: Y1 = {seq1.k where k is Nat : n <= k} and
A31: (superior_realsequence seq1).n = sup Y1 by RINFSUP2:def 7;
    now
      let w be object;
A32:  r in {r} by TARSKI:def 1;
      assume w in Y1;
      then consider k1 be Nat such that
A33:  w = seq1.k1 and
A34:  n <= k1 by A30;
      reconsider w1=w as R_eal by A33;
A35:  seq2.k1 in Y2 by A28,A34;
      w1 = r + seq2.k1 by A2,A33;
      hence w in {r} + Y2 by A32,A35;
    end;
    then
A36: Y1 c= {r} + Y2 by TARSKI:def 3;
    now
      let w be object;
      assume w in {r} + Y2;
      then consider w1,w2 be R_eal such that
A37:  w = w1 + w2 and
A38:  w1 in {r} and
A39:  w2 in Y2;
      consider k2 be Nat such that
A40:  w2 = seq2.k2 and
A41:  n <= k2 by A28,A39;
      w1 = r by A38,TARSKI:def 1;
      then w = seq1.k2 by A2,A37,A40;
      hence w in Y1 by A30,A41;
    end;
    then {r} + Y2 c= Y1 by TARSKI:def 3;
    then Y1 = {r} + Y2 by A36,XBOOLE_0:def 10;
    then sup Y1 = sup{r} + sup Y2 by A1,Th8;
    then sup Y1 = r + sup Y2 by XXREAL_2:11;
    then z = (superior_realsequence seq1).n by A24,A23,A29,A31,TARSKI:def 1;
    hence z in P1 by FUNCT_2:4;
  end;
  then
A42: {r} + P2 c= P1 by TARSKI:def 3;
  now
    let z be object;
    assume z in R1;
    then consider n be object such that
A43: n in NAT and
A44: (inferior_realsequence seq1).n = z by FUNCT_2:11;
    reconsider n as Element of NAT by A43;
    consider Y1 be non empty Subset of ExtREAL such that
A45: Y1 = {seq1.k where k is Nat : n <= k} and
A46: z = inf Y1 by A44,RINFSUP2:def 6;
    consider Y2 be non empty Subset of ExtREAL such that
A47: Y2 = {seq2.k where k is Nat : n <= k} and
A48: (inferior_realsequence seq2).n = inf Y2 by RINFSUP2:def 6;
    now
      let w be object;
A49:  r in {r} by TARSKI:def 1;
      assume w in Y1;
      then consider k1 be Nat such that
A50:  w = seq1.k1 and
A51:  n <= k1 by A45;
      reconsider w1=w as R_eal by A50;
A52:  seq2.k1 in Y2 by A47,A51;
      w1 = r + seq2.k1 by A2,A50;
      hence w in {r} + Y2 by A49,A52;
    end;
    then
A53: Y1 c= {r} + Y2 by TARSKI:def 3;
    now
      let w be object;
      assume w in {r} + Y2;
      then consider w1,w2 be R_eal such that
A54:  w = w1 + w2 and
A55:  w1 in {r} and
A56:  w2 in Y2;
      consider k2 be Nat such that
A57:  w2 = seq2.k2 and
A58:  n <= k2 by A47,A56;
      w1 = r by A55,TARSKI:def 1;
      then w = seq1.k2 by A2,A54,A57;
      hence w in Y1 by A45,A58;
    end;
    then {r} + Y2 c= Y1 by TARSKI:def 3;
    then Y1 = {r} + Y2 by A53,XBOOLE_0:def 10;
    then inf Y1 = inf{r} + inf Y2 by A1,Th9;
    then
A59: inf Y1 = r + inf Y2 by XXREAL_2:13;
A60: (inferior_realsequence seq2).n in R2 by FUNCT_2:4;
    r in {r} by TARSKI:def 1;
    hence z in {r} + R2 by A46,A48,A59,A60;
  end;
  then R1 c= {r} + R2 by TARSKI:def 3;
  then R1 = {r} + R2 by A22,XBOOLE_0:def 10;
  then sup R1 = sup {r} + sup R2 by A1,Th8;
  hence lim_inf seq1 = r + lim_inf seq2 by XXREAL_2:11;
  now
    let z be object;
    assume z in P1;
    then consider n be object such that
A61: n in NAT and
A62: (superior_realsequence seq1).n = z by FUNCT_2:11;
    reconsider n as Element of NAT by A61;
    consider Y1 be non empty Subset of ExtREAL such that
A63: Y1 = {seq1.k where k is Nat : n <= k} and
A64: z = sup Y1 by A62,RINFSUP2:def 7;
    consider Y2 be non empty Subset of ExtREAL such that
A65: Y2 = {seq2.k where k is Nat : n <= k} and
A66: (superior_realsequence seq2).n = sup Y2 by RINFSUP2:def 7;
    now
      let w be object;
A67:  r in {r} by TARSKI:def 1;
      assume w in Y1;
      then consider k1 be Nat such that
A68:  w = seq1.k1 and
A69:  n <= k1 by A63;
      reconsider w1=w as R_eal by A68;
A70:  seq2.k1 in Y2 by A65,A69;
      w1 = r + seq2.k1 by A2,A68;
      hence w in {r} + Y2 by A67,A70;
    end;
    then
A71: Y1 c= {r} + Y2 by TARSKI:def 3;
    now
      let w be object;
      assume w in {r} + Y2;
      then consider w1,w2 be R_eal such that
A72:  w = w1 + w2 and
A73:  w1 in {r} and
A74:  w2 in Y2;
      consider k2 be Nat such that
A75:  w2 = seq2.k2 and
A76:  n <= k2 by A65,A74;
      w1 = r by A73,TARSKI:def 1;
      then w = seq1.k2 by A2,A72,A75;
      hence w in Y1 by A63,A76;
    end;
    then {r} + Y2 c= Y1 by TARSKI:def 3;
    then Y1 = {r} + Y2 by A71,XBOOLE_0:def 10;
    then sup Y1 = sup{r} + sup Y2 by A1,Th8;
    then
A77: sup Y1 = r + sup Y2 by XXREAL_2:11;
A78: (superior_realsequence seq2).n in P2 by FUNCT_2:4;
    r in {r} by TARSKI:def 1;
    hence z in {r} + P2 by A64,A66,A77,A78;
  end;
  then P1 c= {r} + P2 by TARSKI:def 3;
  then P1 = {r} + P2 by A42,XBOOLE_0:def 10;
  then inf P1 = inf{r} + inf P2 by A1,Th9;
  hence thesis by XXREAL_2:13;
end;
