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 Th14:
  (seq is divergent_to-infty & r>0 implies r(#)seq is
  divergent_to-infty) & (seq is divergent_to-infty & r<0 implies r(#)seq is
  divergent_to+infty) & ( r=0 implies rng (r(#)seq)={0} & r(#) seq is constant)
proof
  thus seq is divergent_to-infty & r>0 implies r(#)seq is divergent_to-infty
  proof
    assume that
A1: seq is divergent_to-infty and
A2: r>0;
    let g;
    consider n such that
A3: for m st n<=m holds seq.m<g/r by A1;
    take n;
    let m;
    assume n<=m;
    then seq.m<g/r by A3;
    then r*seq.m<g/r*r by A2,XREAL_1:68;
    then r*seq.m<g by A2,XCMPLX_1:87;
    hence thesis by SEQ_1:9;
  end;
  thus seq is divergent_to-infty & r<0 implies r(#)seq is divergent_to+infty
  proof
    assume that
A4: seq is divergent_to-infty and
A5: r<0;
    let g;
    consider n such that
A6: for m st n<=m holds seq.m<g/r by A4;
    take n;
    let m;
    assume n<=m;
    then seq.m<g/r by A6;
    then g/r*r<r*seq.m by A5,XREAL_1:69;
    then g<r*seq.m by A5,XCMPLX_1:87;
    hence thesis by SEQ_1:9;
  end;
  assume
A7: r=0;
  thus rng(r(#)seq)={0}
  proof
    let x be Real;
    thus x in rng(r(#)seq) implies x in {0}
    proof
      assume x in rng(r(#)seq);
      then consider n being Element of NAT such that
A8:   x=(r(#)seq).n by FUNCT_2:113;
      x=r*seq.n by A8,SEQ_1:9
        .=0 by A7;
      hence thesis by TARSKI:def 1;
    end;
    assume x in {0};
    then
A9: x=0 by TARSKI:def 1;
    (r(#)seq).0=r*seq.0 by SEQ_1:9
      .=0 by A7;
    hence thesis by A9,VALUED_0:28;
  end;
  hence thesis by FUNCT_2:111,Lm4;
end;
