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 Th43:
  seq is divergent_to-infty & (for n holds seq1.n<=seq.n) implies
  seq1 is divergent_to-infty
proof
  assume that
A1: seq is divergent_to-infty and
A2: for n holds seq1.n<= seq.n;
  let r;
  consider n such that
A3: for m st n<=m holds seq.m<r by A1;
  take n;
  let m;
  assume n<=m;
  then
A4: seq.m<r by A3;
  seq1.m<=seq.m by A2;
  hence thesis by A4,XXREAL_0:2;
end;
