reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th14:
  for s being convergent Complex_Sequence, r being Complex
   holds lim(r(#)s)=r*(lim s)
proof
  let s being convergent Complex_Sequence, r be Complex;
  reconsider r as Element of COMPLEX by XCMPLX_0:def 2;
  per cases;
   suppose
A1: r<>0c;
    for p be Real st p>0 holds ex n st for m st n<=m holds |.(r(#)s)
    .m-r*(lim s).| <p
    proof
      let p be Real such that
A2:   p>0;
A3:   |.r.|>0 by A1,COMPLEX1:47;
      p / |.r.| > 0 by A3,A2;
      then consider n such that
A4:   for m st n<=m holds |.s.m-(lim s).|< p/|.r.| by Def6;
      take n;
      let m;
      assume n<=m;
      then |.s.m-(lim s).|<p/|.r.| by A4;
      then |.s.m-(lim s).|*|.r.|<p / |.r.|*|.r.| by A3,XREAL_1:68;
      then |.s.m-(lim s).|*|.r.|< p * (|.r.|)"*|.r.| by XCMPLX_0:def 9;
      then |.s.m-(lim s).|*|.r.|< p * ((|.r.|)"*|.r.|);
      then
A5:   |.s.m-(lim s).|*|.r.|< p * 1 by A3,XCMPLX_0:def 7;
      |.(r(#)s).m - r*(lim s).|=|.r*s.m - r*(lim s).| by VALUED_1:6
        .= |. r *(s.m-(lim s)).|
        .= |.s.m-(lim s).|*|.r.| by COMPLEX1:65;
      hence thesis by A5;
    end;
    hence thesis by Def6;
   end;
  suppose
A6: r=0c;
    now
      let n;
      thus (r(#)s).n = 0c*s.n by A6,VALUED_1:6
        .= 0c;
    end;
    hence thesis by A6,Th10;
   end;
end;
