reserve X for 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 Th5:
  s is convergent implies s*z is convergent
proof
A1: 0<=||.z.|| by NORMSP_1:4;
  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=g1*z;
  let p be Real;
A3: 0+0< ||.z.||+1 by NORMSP_1:4,XREAL_1:8;
  assume
A4: 0<p;
  then consider n such that
A5: for m st n<=m holds ||.s.m-g1.||<p/(||.z.||+1) by A2,A3,XREAL_1:139;
  take n;
  let m;
  assume n<=m;
  then
A6: ||.s.m-g1.||< p/(||.z.||+1) by A5;
A7: ||.(s.m-g1)*z.|| <=||.s.m-g1.||* ||.z.|| by LOPBAN_3:38;
A8: 0+ ||.z.|| < ||.z.||+1 by XREAL_1:8;
  0<p/(||.z.||+1) by A3,A4,XREAL_1:139;
  then
A9: (p/(||.z.||+1))*||.z.||< (p/(||.z.||+1)) * ( ||.z.||+1) by A1,A8,XREAL_1:97
;
A10: ||.((s*z).m)-g.|| =||.s.m*z-g1*z.|| by LOPBAN_3:def 6
    .=||.(s.m-g1)*z.|| by LOPBAN_3:38;
  0<=||.s.m-g1.|| by NORMSP_1:4;
  then ||.s.m-g1.||* ||.z.||<=(p/(||.z.||+1))* ||.z.|| by A1,A6,XREAL_1:66;
  then
A11: ||.(s.m-g1)*z.||<=(p/(||.z.||+1))*||.z.|| by A7,XXREAL_0:2;
  (p/(||.z.||+1))* ( ||.z.||+1) =p by A3,XCMPLX_1:87;
  hence ||.((s*z).m)-g.|| < p by A10,A11,A9,XXREAL_0:2;
end;
