
theorem
  for seq be Real_Sequence, r be Real st seq is bounded & 0<=r holds
  lim_sup(r(#)seq)=r*(lim_sup seq)
proof
  let seq be Real_Sequence, r be Real;
  assume that
A1: seq is bounded and
A2: 0<=r;
  superior_realsequence seq in Funcs(NAT,REAL) by FUNCT_2:8;
  then
A3: ex f be Function st superior_realsequence seq = f & dom f = NAT & rng f
  c= REAL by FUNCT_2:def 2;
  (superior_realsequence seq).0 in rng superior_realsequence seq by FUNCT_2:4;
  then reconsider
  X1 = rng superior_realsequence seq as non empty Subset of REAL by A3;
  now
    let n be Element of NAT;
    consider r1 be Real such that
A4: 0<r1 and
A5: for k be Nat holds |.seq.k.|<r1 by A1,SEQ_2:3;
    seq ^\n in Funcs(NAT,REAL) by FUNCT_2:8;
    then ex f be Function st seq ^\n = f & dom f = NAT & rng f c= REAL by
FUNCT_2:def 2;
    then reconsider Y1 = rng (seq ^\n) as non empty Subset of REAL
           by FUNCT_2:4;
    now
      let k be Nat;
      |.(seq ^\n).k.| = |.seq.(n+k).| by NAT_1:def 3;
      hence |.(seq ^\n).k.| < r1 by A5;
    end;
    then seq ^\n is bounded by A4,SEQ_2:3;
    then
A6: Y1 is bounded_above by RINFSUP1:5;
    (superior_realsequence (r (#) seq)).n = upper_bound ((r (#) seq) ^\n) by
RINFSUP1:37
      .= upper_bound (r (#) (seq^\n)) by SEQM_3:21
      .= upper_bound (r ** Y1) by INTEGRA2:17
      .= r * upper_bound (seq ^\n) by A2,A6,INTEGRA2:13
      .= r * (superior_realsequence seq).n by RINFSUP1:37;
    hence
    (superior_realsequence (r (#) seq)).n = (r (#) (superior_realsequence
    seq)).n by SEQ_1:9;
  end;
  then superior_realsequence (r (#) seq) = r (#) (superior_realsequence seq)
  by FUNCT_2:63;
  then
A7: rng superior_realsequence (r (#) seq) = r ** X1 by INTEGRA2:17;
  superior_realsequence seq is bounded by A1,RINFSUP1:56;
  then X1 is bounded_below by RINFSUP1:6;
  hence thesis by A2,A7,INTEGRA2:15;
end;
