reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th16:
  seq1 is bounded_above & seq2 is bounded_above implies upper_bound (seq1
  + seq2) <= upper_bound seq1 + upper_bound seq2
proof
  assume that
A1: seq1 is bounded_above and
A2: seq2 is bounded_above;
  for n holds (seq1 + seq2).n <= upper_bound seq1 + upper_bound seq2
  proof
    let n;
A3: seq2.n <= upper_bound seq2 by A2,Th7;
    (seq1 + seq2).n = seq1.n + seq2.n & seq1.n <= upper_bound seq1
    by A1,Th7,SEQ_1:7;
    hence thesis by A3,XREAL_1:7;
  end;
  hence thesis by Th9;
end;
