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 iff -seq is bounded_below
proof
  set seq1=-seq;
  thus seq is bounded_above implies -seq is bounded_below;
  assume seq1 is bounded_below;
  then consider t such that
A1: for n holds seq1.n>t by SEQ_2:def 4;
  for n holds seq.n < -t
  proof
    let n;
    seq1.n=-seq.n by SEQ_1:10;
    then
A2: -seq1.n = seq.n;
    seq1.n > t by A1;
    hence thesis by A2,XREAL_1:24;
  end;
  hence thesis by SEQ_2:def 3;
end;
