reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;

theorem Th36:
  (seq is bounded_above & 0<r implies r(#)seq is bounded_above) &
(0=r implies r(#)seq is bounded) & (seq is bounded_above & r<0 implies r(#)seq
  is bounded_below)
proof
  thus seq is bounded_above & 0<r implies r(#)seq is bounded_above;
  thus 0=r implies r(#)seq is bounded
  proof
    assume
A1: 0=r;
    now
      let n;
      (r(#)seq).n = 0*seq.n by A1,SEQ_1:9;
      hence (r(#)seq).n < 1;
    end;
    hence r(#)seq is bounded_above;
    take p=-1;
    let n be Nat;
    -1<0 & r*seq.n=0 by A1;
    hence p<(r(#)seq).n by SEQ_1:9;
  end;
  assume that
A2: seq is bounded_above and
A3: r<0;
  consider r1 such that
A4: for n holds seq.n<r1 by A2;
  take p=r*|.r1.|;
  let n be Nat;
  r1<=|.r1.| by ABSVALUE:4;
  then seq.n<|.r1.| by A4,XXREAL_0:2;
  then r*|.r1.|<r*seq.n by A3,XREAL_1:69;
  hence p<(r(#)seq).n by SEQ_1:9;
end;
