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 Th4:
  s is convergent implies z * s is convergent
proof
A1: 0<=||.z.|| by CLVECT_1:105;
  assume s is convergent;
  then consider g1 be Point of X such that
A2: for p being Real st 0<p ex n st for m st n<=m holds ||.s.m-g1.||<p;
  take g=z*g1;
  let p be Real;
  assume
A3: 0<p;
A4: 0+0< ||.z.||+1 by CLVECT_1:105,XREAL_1:8;
  then consider n such that
A5: for m st n<=m holds ||.s.m-g1.||<p/(||.z.||+1) by A2,A3;
  take n;
  let m;
  assume n<=m;
  then
A6: ||.s.m-g1.||< p/(||.z.||+1) by A5;
A7: ||.z*(s.m-g1).|| <= ||.z.||*||.s.m-g1.|| by CLOPBAN3:38;
  0<=||.s.m-g1.|| by CLVECT_1:105;
  then ||.z.||*||.s.m-g1.||<= ||.z.||*(p/(||.z.||+1)) by A1,A6,XREAL_1:66;
  then
A8: ||.z*(s.m-g1).||<=||.z.||*(p/(||.z.||+1)) by A7,XXREAL_0:2;
A9: ||.((z*s).m)-g.|| =||.z*s.m-z*g1.|| by LOPBAN_3:def 5
    .=||.z*(s.m-g1).|| by CLOPBAN3:38;
  0+ ||.z.|| < ||.z.||+1 by XREAL_1:8;
  then
A10: ||.z.||*(p/(||.z.||+1)) < ( ||.z.||+1) *(p/(||.z.||+1)) by A1,A3,
XREAL_1:97;
  ( ||.z.||+1) *(p/(||.z.||+1)) =p by A4,XCMPLX_1:87;
  hence thesis by A9,A8,A10,XXREAL_0:2;
end;
