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_below & seq2 is bounded_below implies (
  inferior_realsequence(seq1+seq2)).n >= (inferior_realsequence seq1).n + (
  inferior_realsequence seq2).n
proof
  assume seq1 is bounded_below & seq2 is bounded_below;
  then seq1 ^\n is bounded_below & seq2 ^\n is bounded_below by SEQM_3:28;
  then lower_bound(seq1 ^\n + seq2 ^\n) >=
  lower_bound(seq1 ^\n) + lower_bound(seq2 ^\n) by Th15;
  then
A1: lower_bound((seq1+seq2) ^\n) >=
lower_bound(seq1 ^\n) + lower_bound(seq2 ^\n) by SEQM_3:15;
  (inferior_realsequence seq1).n =
  lower_bound(seq1 ^\n) & (inferior_realsequence
  seq2 ).n = lower_bound(seq2 ^\n) by Th36;
  hence thesis by A1,Th36;
end;
