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 Th12:
  seq1 is divergent_to-infty & seq2 is bounded_above implies seq1+
  seq2 is divergent_to-infty
proof
  assume that
A1: seq1 is divergent_to-infty and
A2: seq2 is bounded_above;
  let r;
  consider M be Real such that
A3: for n holds seq2.n<M by A2;
  consider n such that
A4: for m st n<=m holds seq1.m<r-M by A1;
  take n;
  let m;
  assume n<=m;
  then seq1.m<r-M by A4;
  then seq1.m+seq2.m<r-M+M by A3,XREAL_1:8;
  hence thesis by SEQ_1:7;
end;
