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 nonnegative & seq2 is bounded_below nonnegative
implies (inferior_realsequence(seq1(#)seq2)).n >= (inferior_realsequence seq1).
  n * (inferior_realsequence seq2).n
proof
  assume seq1 is bounded_below nonnegative & seq2 is bounded_below nonnegative;
  then seq1 ^\n is bounded_below nonnegative & seq2 ^\n is bounded_below
  nonnegative by Th17,SEQM_3:28;
  then lower_bound((seq1 ^\n)(#)(seq2 ^\n))>=
  lower_bound(seq1 ^\n) * lower_bound(seq2 ^\n) by Th20;
  then
A1: lower_bound((seq1 (#) seq2) ^\n) >=
lower_bound(seq1 ^\n) * lower_bound(seq2 ^\n) by SEQM_3:19;
  (inferior_realsequence seq1).n = lower_bound(seq1 ^\n) &
  (inferior_realsequence
  seq2 ).n = lower_bound(seq2 ^\n) by Th36;
  hence thesis by A1,Th36;
end;
