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 Th80:
  seq1 is bounded_above nonnegative & seq2 is bounded_above
  nonnegative implies (superior_realsequence(seq1(#)seq2)).n <= (
  superior_realsequence seq1).n * (superior_realsequence seq2).n
proof
  assume seq1 is bounded_above nonnegative & seq2 is bounded_above nonnegative;
  then seq1 ^\n is bounded_above nonnegative & seq2 ^\n is bounded_above
  nonnegative by Th17,SEQM_3:27;
  then upper_bound((seq1 ^\n)(#)(seq2 ^\n))<=
  upper_bound(seq1 ^\n) * upper_bound(seq2 ^\n) by Th21;
  then
A1: upper_bound((seq1 (#) seq2) ^\n) <=
upper_bound(seq1 ^\n) * upper_bound(seq2 ^\n) by SEQM_3:19;
  (superior_realsequence seq1).n = upper_bound(seq1 ^\n) &
  (superior_realsequence
  seq2 ).n = upper_bound(seq2 ^\n) by Th37;
  hence thesis by A1,Th37;
end;
