
theorem Th21:
for seq1,seq2 be without-infty ExtREAL_sequence
 st seq1 is convergent_to_-infty & seq2 is convergent_to_-infty
  holds seq1 + seq2 is convergent_to_-infty & seq1 + seq2 is convergent
      & lim(seq1+seq2) = -infty
proof
   let seq1,seq2 be without-infty ExtREAL_sequence;
   assume A1: seq1 is convergent_to_-infty & seq2 is convergent_to_-infty;
   now let g be Real;
    assume A2: g < 0; then
    consider n1 be Nat such that
A3:  for m be Nat st n1<=m holds seq1.m <= g/2 by A1,MESFUNC5:def 10;
    consider n2 be Nat such that
A4:  for m be Nat st n2<=m holds seq2.m <= g/2 by A1,A2,MESFUNC5:def 10;
    reconsider N1=n1, N2=n2 as Element of NAT by ORDINAL1:def 12;
    reconsider n = max(N1,N2) as Nat;
A5: n1<=n & n2<=n by XXREAL_0:25;
    now let m be Nat;
     assume n<=m; then
     n1<=m & n2<=m by A5,XXREAL_0:2; then
     seq1.m <= g/2 & seq2.m <= g/2 by A3,A4; then
A6:  seq1.m + seq2.m <= (g/2) qua ExtReal + g/2 by XXREAL_3:36;
     m is Element of NAT by ORDINAL1:def 12;
     hence (seq1+seq2).m <= g by A6,Th7;
    end;
    hence ex n be Nat st for m be Nat st n<=m holds (seq1+seq2).m <= g;
   end;
   hence
A7: seq1+seq2 is convergent_to_-infty by MESFUNC5:def 10;
   hence seq1+seq2 is convergent;
   thus lim(seq1+seq2)=-infty by A7,MESFUNC5:def 12;
end;
