
theorem Th22:
for seq1,seq2 be without-infty ExtREAL_sequence
 st seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number
  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_finite_number; then
    consider S2 be Real such that
A2:  for g be Real st 0<g ex n be Nat st for m be Nat st n<=m
      holds |.seq2.m - S2 qua ExtReal.| < g by MESFUNC5:def 8;
   now let g be Real;
    assume A3: g < 0;
    set G = min(-1,2*g - S2);
A4: G <= -1 & G <= 2*g - S2 by XXREAL_0:17; then
    consider n1 be Nat such that
A5:  for m be Nat st n1<=m holds seq1.m<=G by A1,MESFUNC5:def 10;
    consider n2 be Nat such that
A6:  for m be Nat st n2<=m holds |.seq2.m - S2 qua ExtReal.| < -g by A2,A3;
    reconsider N1=n1, N2=n2 as Element of NAT by ORDINAL1:def 12;
    reconsider n = max(N1,N2) as Nat;
A7: n1<=n & n2<=n by XXREAL_0:25;
    now let m be Nat;
     assume n<=m; then
     n1<=m & n2<=m by A7,XXREAL_0:2; then
A8:  seq1.m <= G & |.seq2.m - S2 qua ExtReal.| < -g by A5,A6;
     reconsider g1=g as R_eal by XXREAL_0:def 1;
B1:  -g1 = -g by XXREAL_3:def 3; then
     seq2.m - S2 qua ExtReal < -g1 by A8,EXTREAL1:21; then
     seq2.m < -g1 + (S2 qua ExtReal) by XXREAL_3:54; then
A9:  seq1.m + seq2.m <= G + (-g1 + (S2 qua ExtReal)) by A8,XXREAL_3:36;
     (-g1 + (S2 qua ExtReal)) = -g + S2 by B1,XXREAL_3:def 2; then
     2*g - S2 + (-g1 + (S2 qua ExtReal))
      = 2*g - S2 + (-g + S2) by XXREAL_3:def 2; then
     G + (-g1 + (S2 qua ExtReal)) <= g by A4,XXREAL_3:36; then
A10: seq1.m + seq2.m <= g by A9,XXREAL_0:2;
     m is Element of NAT by ORDINAL1:def 12;
     hence (seq1+seq2).m <= g by A10,Th7;
    end;
    hence ex n be Nat st for m be Nat st n<=m holds (seq1+seq2).m <= g;
   end;
   hence
A11:seq1+seq2 is convergent_to_-infty by MESFUNC5:def 10;
   hence seq1+seq2 is convergent;
   thus lim(seq1+seq2)=-infty by A11,MESFUNC5:def 12;
end;
