theorem Th15:
  seq1 is bounded_below & seq2 is bounded_below implies lower_bound (seq1
  + seq2) >= lower_bound seq1 + lower_bound seq2
proof
  assume that
A1: seq1 is bounded_below and
A2: seq2 is bounded_below;
  for n holds (seq1 + seq2).n >= lower_bound seq1 + lower_bound seq2
  proof
    let n;
A3: seq2.n >= lower_bound seq2 by A2,Th8;
    (seq1 + seq2).n = seq1.n + seq2.n & seq1.n >= lower_bound seq1
    by A1,Th8,SEQ_1:7;
    hence thesis by A3,XREAL_1:7;
  end;
  hence thesis by Th10;
end;
