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
  seq1 is bounded_above & seq2 is bounded_above implies (
  superior_realsequence(seq1+seq2)).n <= (superior_realsequence seq1).n + (
  superior_realsequence seq2).n
proof
  assume seq1 is bounded_above & seq2 is bounded_above;
  then seq1 ^\n is bounded_above & seq2 ^\n is bounded_above by SEQM_3:27;
  then upper_bound(seq1 ^\n + seq2 ^\n) <=
  upper_bound(seq1 ^\n) + upper_bound(seq2 ^\n) by Th16;
  then
A1: upper_bound((seq1+seq2) ^\n) <=
upper_bound(seq1 ^\n) + upper_bound(seq2 ^\n) by SEQM_3:15;
  (superior_realsequence seq1).n = upper_bound(seq1 ^\n) &
  (superior_realsequence
  seq2 ).n = upper_bound(seq2 ^\n) by Th37;
  hence thesis by A1,Th37;
end;
