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 non-decreasing & 0<=r implies r(#)seq is non-decreasing) & (
  seq is non-decreasing & r<=0 implies r(#)seq is non-increasing)
proof
  thus seq is non-decreasing & 0<=r implies r(#)seq is non-decreasing
  proof
    assume that
A1: seq is non-decreasing and
A2: 0<=r;
    let n;
    seq.n<=seq.(n+1) by A1;
    then r*seq.n<=r*seq.(n+1) by A2,XREAL_1:64;
    then (r(#)seq).n<=r*seq.(n+1) by SEQ_1:9;
    hence thesis by SEQ_1:9;
  end;
  assume that
A3: seq is non-decreasing and
A4: r<=0;
  let n;
  seq.n<=seq.(n+1) by A3;
  then r*seq.(n+1)<=r*seq.n by A4,XREAL_1:65;
  then (r(#)seq).(n+1)<=r*seq.n by SEQ_1:9;
  hence thesis by SEQ_1:9;
end;
