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 Th3:
  s is convergent & s9 is convergent implies s * s9 is convergent
proof
  assume that
A1: s is convergent and
A2: s9 is convergent;
  consider g1 be Point of X such that
A3: for p being Real st 0<p ex n st for m st n<=m holds ||.s.m-g1.||<p
  by A1;
  ||.s.|| is bounded by A1,NORMSP_1:23,SEQ_2:13;
  then consider R be Real such that
A4: for n being Nat holds ||.s.||.n < R by SEQ_2:def 3;
A5: now
    let n;
    ||.s.n.||= ||.s.||.n by NORMSP_0:def 4;
    hence ||.s.n.|| < R by A4;
  end;
  ||. s.1 .|| = ||. s .||.1 by NORMSP_0:def 4;
  then 0<= ||.s.||.1 by NORMSP_1:4;
  then
A6: 0 < R by A4;
  consider g2 be Point of X such that
A7: for p being Real st 0<p ex n st for m st n<=m holds ||.s9.m-g2.||<p
  by A2;
  take g=g1*g2;
  let p be Real;
  reconsider R as Real;
A8: 0+0<||.g2.||+R by A6,NORMSP_1:4,XREAL_1:8;
  assume
A9: 0<p;
  then consider n1 such that
A10: for m st n1<=m holds ||.s.m-g1.||<p/(||.g2.||+R) by A3,A8,XREAL_1:139;
  consider n2 such that
A11: for m st n2<=m holds ||.s9.m-g2.||<p/(||.g2.||+R) by A7,A8,A9,XREAL_1:139;
  take n=n1+n2;
  let m such that
A12: n<=m;
  n2<=n by NAT_1:12;
  then n2<=m by A12,XXREAL_0:2;
  then
A13: ||.s9.m-g2.||<p/(||.g2.||+R) by A11;
A14: 0<=||.s.m.|| by NORMSP_1:4;
A15: ||.s.m*(s9.m-g2).|| <= ||.s.m.||*||.s9.m-g2.|| by LOPBAN_3:38;
A16: 0<=||.s9.m-g2.|| by NORMSP_1:4;
  n1<=n1+n2 by NAT_1:12;
  then n1<=m by A12,XXREAL_0:2;
  then
A17: ||.s.m-g1.||<=p/(||.g2.||+R) by A10;
  ||.((s*s9).m)-g.|| =||.s.m*s9.m-g1*g2.|| by LOPBAN_3:def 7
    .=||.(s.m*s9.m-s.m*g2)+(s.m*g2-g1*g2).|| by LOPBAN_3:38
    .=||.s.m*(s9.m-g2)+(s.m*g2-g1*g2).|| by LOPBAN_3:38
    .=||.s.m*(s9.m-g2)+(s.m-g1)*g2.|| by LOPBAN_3:38;
  then
A18: ||.((s*s9).m)-g.||<= ||.s.m*(s9.m-g2).||+||.(s.m-g1)*g2.|| by
NORMSP_1:def 1;
  ||.s.m.||<R by A5;
  then ||.s.m.||*||.s9.m-g2.||<R*(p/(||.g2.||+R)) by A14,A16,A13,XREAL_1:96;
  then
A19: ||.s.m*(s9.m-g2).||<R*(p/(||.g2.||+R)) by A15,XXREAL_0:2;
A20: ||.(s.m-g1)*g2.||<=||.g2.||*||.s.m-g1.|| by LOPBAN_3:38;
  0<=||.g2.|| by NORMSP_1:4;
  then ||.g2.||*||.s.m-g1.|| <=||.g2.||*(p/(||.g2.||+R)) by A17,XREAL_1:64;
  then
A21: ||.(s.m-g1)*g2.||<=||.g2.||*(p/(||.g2.||+R)) by A20,XXREAL_0:2;
  R*(p/(||.g2.||+R))+||.g2.||*(p/(||.g2.||+R)) =(p/(||.g2.||+R))*(||.g2 .||+R)
    .=p by A8,XCMPLX_1:87;
  then ||.s.m*(s9.m-g2).||+||.(s.m-g1)*g2.||<p by A19,A21,XREAL_1:8;
  hence ||.((s*s9).m)-g.||<p by A18,XXREAL_0:2;
end;
