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 increasing & 0<r implies r(#)seq is increasing) & (0=r implies
r(#)seq is constant) & (seq is increasing & r<0 implies r(#)seq is decreasing)
proof
  thus seq is increasing & 0<r implies r(#)seq is increasing
  proof
    assume that
A1: seq is increasing and
A2: 0<r;
    let n;
    seq.n<seq.(n+1) by A1;
    then r*seq.n<r*seq.(n+1) by A2,XREAL_1:68;
    then (r(#)seq).n<r*seq.(n+1) by SEQ_1:9;
    hence thesis by SEQ_1:9;
  end;
  thus 0=r implies r(#)seq is constant
  proof
    assume
A3: 0=r;
    now
      let n be Nat;
      n in NAT & r*seq.n=r*seq.(n+1) by A3,ORDINAL1:def 12;
      then (r(#)seq).n=r*seq.(n+1) by SEQ_1:9;
      hence (r(#)seq).n=(r(#)seq).(n+1) by SEQ_1:9;
    end;
    hence thesis by VALUED_0:25;
  end;
  assume that
A4: seq is increasing and
A5: r<0;
  let n;
  seq.n<seq.(n+1) by A4;
  then r*seq.(n+1)<r*seq.n by A5,XREAL_1:69;
  then (r(#)seq).(n+1)<r*seq.n by SEQ_1:9;
  hence thesis by SEQ_1:9;
end;
