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 nonnegative & seq2 is bounded nonnegative implies (
  lim_inf seq1) * (lim_inf seq2) <= lim_inf(seq1(#)seq2) & lim_sup(seq1(#)seq2)
  <= (lim_sup seq1) * (lim_sup seq2)
proof
A1: for seq1,seq2 st seq1 is bounded & seq1 is nonnegative & seq2 is
bounded & seq2 is nonnegative holds lim_sup(seq1(#)seq2) <= (lim_sup seq1) * (
  lim_sup seq2)
  proof
    let seq1,seq2;
    assume that
A2: seq1 is bounded nonnegative and
A3: seq2 is bounded nonnegative;
    set seq3 = superior_realsequence seq1, seq4 = superior_realsequence seq2,
    seq5 = superior_realsequence(seq1(#)seq2);
A4: seq5 is convergent by A2,A3,Th58;
A5: lower_bound seq5 = lim seq5 & lower_bound seq3 = lim seq3 by A2,A3,Th58;
A6: lower_bound seq4 =lim seq4 by A3,Th58;
A7: for n holds seq5.n <= (seq3(#)seq4).n
    proof
      let n;
      seq5.n <= seq3.n * seq4.n by A2,A3,Th80;
      hence thesis by SEQ_1:8;
    end;
A8: seq3 is convergent & seq4 is convergent by A2,A3,Th58;
    then (seq3(#)seq4) is convergent;
    then lim seq5 <= lim (seq3(#)seq4) by A4,A7,SEQ_2:18;
    hence thesis by A8,A5,A6,SEQ_2:15;
  end;
  for seq1,seq2 st seq1 is bounded nonnegative & seq2 is bounded
  nonnegative holds (lim_inf seq1) * (lim_inf seq2) <= lim_inf(seq1(#)seq2)
  proof
    let seq1,seq2;
    assume that
A9: seq1 is bounded nonnegative and
A10: seq2 is bounded nonnegative;
    set seq3 = inferior_realsequence seq1, seq4 = inferior_realsequence seq2,
    seq5 = inferior_realsequence(seq1(#)seq2);
A11: seq5 is convergent by A9,A10,Th57;
A12: upper_bound seq5 = lim seq5 & upper_bound seq3 = lim seq3 by A9,A10,Th57;
A13: upper_bound seq4 =lim seq4 by A10,Th57;
A14: for n holds seq5.n >= (seq3(#)seq4).n
    proof
      let n;
      seq5.n >= seq3.n * seq4.n by A9,A10,Th79;
      hence thesis by SEQ_1:8;
    end;
A15: seq3 is convergent & seq4 is convergent by A9,A10,Th57;
    then (seq3(#)seq4) is convergent;
    then lim seq5 >= lim (seq3(#)seq4) by A11,A14,SEQ_2:18;
    hence thesis by A15,A12,A13,SEQ_2:15;
  end;
  hence thesis by A1;
end;
