reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem Th11:
  seq1 is divergent_to-infty & seq2 is divergent_to-infty implies
  seq1+seq2 is divergent_to-infty
proof
  assume that
A1: seq1 is divergent_to-infty and
A2: seq2 is divergent_to-infty;
  let r;
  consider n1 be Nat such that
A3: for m st n1<=m holds seq1.m<r/2 by A1;
  consider n2 be Nat such that
A4: for m st n2<=m holds seq2.m<r/2 by A2;
  reconsider n=max(n1,n2) as Nat by TARSKI:1;
  take n;
  let m such that
A5: n<=m;
  n2<=n by XXREAL_0:25;
  then n2<=m by A5,XXREAL_0:2;
  then
A6: seq2.m<r/2 by A4;
  n1<=n by XXREAL_0:25;
  then n1<=m by A5,XXREAL_0:2;
  then seq1.m<r/2 by A3;
  then seq1.m+seq2.m<r/2+r/2 by A6,XREAL_1:8;
  hence thesis by SEQ_1:7;
end;
