reserve X for Complex_Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1, n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th7:
  s is convergent implies lim (s*z) = lim(s)*z
proof
  assume
A1: s is convergent;
  set g1=lim(s);
  set g=g1*z;
A2: 0+0< ||.z.||+1 by CLVECT_1:105,XREAL_1:8;
A3: 0<=||.z.|| by CLVECT_1:105;
A4: now
    let p be Real;
    assume
A5: 0<p;
    then consider n such that
A6: for m st n<=m holds ||.s.m-g1.||<p/(||.z.||+1) by A1,A2,CLVECT_1:def 16;
    take n;
    let m;
    assume n<=m;
    then
A7: ||.s.m-g1.||< p/(||.z.||+1) by A6;
    0<=||.s.m-g1.|| by CLVECT_1:105;
    then
A8: ||.s.m-g1.||* ||.z.||<=(p/(||.z.||+1))* ||.z.|| by A3,A7,XREAL_1:66;
    ||.(s.m-g1)*z.|| <=||.s.m-g1.||* ||.z.|| by CLOPBAN3:38;
    then
A9: ||.(s.m-g1)*z.||<=(p/(||.z.||+1))*||.z.|| by A8,XXREAL_0:2;
A10: ||.((s*z).m)-g.|| =||.s.m*z-g1*z.|| by LOPBAN_3:def 6
      .=||.(s.m-g1)*z.|| by CLOPBAN3:38;
    0+ ||.z.|| < ||.z.||+1 by XREAL_1:8;
    then
A11: (p/(||.z.||+1))*||.z.||< (p/(||.z.||+1)) * ( ||.z.||+1) by A3,A5,
XREAL_1:97;
    (p/(||.z.||+1))* ( ||.z.||+1) =p by A2,XCMPLX_1:87;
    hence ||.((s*z).m)-g.|| < p by A10,A9,A11,XXREAL_0:2;
  end;
  s*z is convergent by A1,Th5;
  hence thesis by A4,CLVECT_1:def 16;
end;
