
theorem Th24:
for seq1,seq2 be without+infty ExtREAL_sequence holds
  (seq1 is convergent_to_+infty & seq2 is convergent_to_+infty implies
     seq1+seq2 is convergent_to_+infty & seq1+seq2 is convergent
   & lim(seq1+seq2) = +infty)
& (seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number implies
     seq1+seq2 is convergent_to_+infty & seq1+seq2 is convergent
   & lim(seq1+seq2) = +infty)
& (seq1 is convergent_to_-infty & seq2 is convergent_to_-infty implies
     seq1+seq2 is convergent_to_-infty & seq1+seq2 is convergent
   & lim(seq1+seq2) = -infty)
& (seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number implies
     seq1+seq2 is convergent_to_-infty & seq1+seq2 is convergent
   & lim(seq1+seq2) = -infty)
& (seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number
   implies seq1+seq2 is convergent_to_finite_number & seq1+seq2 is convergent
         & lim(seq1+seq2) = lim seq1 + lim seq2)
proof
   let seq1,seq2 be without+infty ExtREAL_sequence;
   reconsider f1=-seq1, f2=-seq2 as without-infty ExtREAL_sequence;
   hereby assume
    seq1 is convergent_to_+infty & seq2 is convergent_to_+infty; then
A2: f1 is convergent_to_-infty & f2 is convergent_to_-infty by Th17; then
    reconsider F = f1+f2 as convergent ExtREAL_sequence by Th21;
A3: f1+f2 is convergent_to_-infty & f1+f2 is convergent & lim(f1+f2)=-infty
      by A2,Th21;
    f1+f2 = f1 - seq2 by Th10 .= -(seq1+seq2) by Th9; then
    seq1+seq2 = -F by Th2;
    hence seq1+seq2 is convergent_to_+infty & seq1+seq2 is convergent
     & lim(seq1+seq2) = +infty by A3,Th17,XXREAL_3:5;
   end;
   hereby assume
    seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number; then
A2: f1 is convergent_to_-infty & f2 is convergent_to_finite_number
      by Th17; then
    reconsider F = f1+f2 as convergent ExtREAL_sequence by Th22;
A3: f1+f2 is convergent_to_-infty & f1+f2 is convergent & lim(f1+f2)=-infty
      by A2,Th22;
    f1+f2 = f1 - seq2 by Th10 .= -(seq1+seq2) by Th9; then
    seq1+seq2 = -F by Th2;
    hence seq1+seq2 is convergent_to_+infty & seq1+seq2 is convergent
     & lim(seq1+seq2) = +infty by A3,Th17,XXREAL_3:5;
   end;
   hereby assume
    seq1 is convergent_to_-infty & seq2 is convergent_to_-infty; then
A2: f1 is convergent_to_+infty & f2 is convergent_to_+infty by Th17; then
    reconsider F = f1+f2 as convergent ExtREAL_sequence by Th18;
A3: f1+f2 is convergent_to_+infty & f1+f2 is convergent & lim(f1+f2)=+infty
      by A2,Th18;
    f1+f2 = f1 - seq2 by Th10 .= -(seq1+seq2) by Th9; then
    seq1+seq2 = -F by Th2;
    hence seq1+seq2 is convergent_to_-infty & seq1+seq2 is convergent
     & lim(seq1+seq2) = -infty by A3,Th17,XXREAL_3:6;
   end;
   hereby assume
    seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number; then
A2: f1 is convergent_to_+infty & f2 is convergent_to_finite_number
      by Th17; then
    reconsider F = f1+f2 as convergent ExtREAL_sequence by Th19;
A3: f1+f2 is convergent_to_+infty & f1+f2 is convergent & lim(f1+f2)=+infty
      by A2,Th19;
    f1+f2 = f1 - seq2 by Th10 .= -(seq1+seq2) by Th9; then
    seq1+seq2 = -F by Th2;
    hence seq1+seq2 is convergent_to_-infty & seq1+seq2 is convergent
     & lim(seq1+seq2) = -infty by A3,Th17,XXREAL_3:6;
   end;
   hereby assume
    seq1 is convergent_to_finite_number
  & seq2 is convergent_to_finite_number; then
A2: f1 is convergent_to_finite_number & f2 is convergent_to_finite_number
      by Th17; then
    reconsider F = f1+f2 as convergent ExtREAL_sequence by Th23;
A3: f1+f2 is convergent_to_finite_number & f1+f2 is convergent &
    lim(f1+f2) = lim f1 + lim f2 by A2,Th23;
    f1+f2 = f1 - seq2 by Th10 .= -(seq1+seq2) by Th9; then
A4: seq1+seq2 = -F by Th2; then
A5: lim(seq1+seq2) = -(lim f1 + lim f2) by A3,Th17;
    seq1 = -f1 & seq2 = -f2 by Th2; then
    lim seq1 = -lim f1 & lim seq2 = -lim f2 by A2,Th17;
    hence seq1+seq2 is convergent_to_finite_number & seq1+seq2 is convergent
     & lim(seq1+seq2) = lim seq1 + lim seq2 by A3,A4,A5,Th17,XXREAL_3:9;
   end;
end;
