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
  seq is bounded_above nonnegative implies upper_bound seq >= 0
proof
  assume
A1: seq is bounded_above nonnegative;
  then
A2: seq.0 <= upper_bound seq by Th7;
  0 <= seq.0 by A1;
  hence thesis by A2;
end;
