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
  (seq is decreasing & 0<r implies r(#)seq is decreasing) & (seq is
  decreasing & r<0 implies r(#)seq is increasing)
proof
  thus seq is decreasing & 0<r implies r(#)seq is decreasing
  proof
    assume that
A1: seq is decreasing and
A2: 0<r;
    let n;
    seq.(n+1)<seq.n by A1;
    then r*seq.(n+1)<r*seq.n by A2,XREAL_1:68;
    then (r(#)seq).(n+1)<r*seq.n by SEQ_1:9;
    hence thesis by SEQ_1:9;
  end;
  assume that
A3: seq is decreasing and
A4: r<0;
  let n;
  seq.(n+1)<seq.n by A3;
  then r*seq.n<r*seq.(n+1) by A4,XREAL_1:69;
  then (r(#)seq).n<r*seq.(n+1) by SEQ_1:9;
  hence thesis by SEQ_1:9;
end;
