reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem Th89:
  f is convergent_in-infty implies r(#)f is convergent_in-infty &
  lim_in-infty(r(#)f)=r*(lim_in-infty f)
proof
  assume
A1: f is convergent_in-infty;
A2: now
    let seq;
    assume that
A3: seq is divergent_to-infty and
A4: rng seq c=dom(r(#)f);
A5: rng seq c=dom f by A4,VALUED_1:def 5;
    then
A6: lim(f/*seq)=lim_in-infty f by A1,A3,Def13;
A7: f/*seq is convergent by A1,A3,A5;
    then r(#)(f/*seq) is convergent;
    hence (r(#)f)/*seq is convergent by A5,RFUNCT_2:9;
    thus lim((r(#)f)/*seq)=lim(r(#)(f/*seq)) by A5,RFUNCT_2:9
      .=r*(lim_in-infty f) by A7,A6,SEQ_2:8;
  end;
  for r1 ex g st g<r1 & g in dom(r(#)f)
  proof
    let r1;
    consider g such that
A8: g<r1 & g in dom f by A1;
    take g;
    thus thesis by A8,VALUED_1:def 5;
  end;
  hence r(#)f is convergent_in-infty by A2;
  hence thesis by A2,Def13;
end;
