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 Th6:
  s is convergent implies lim (z * s) =z* lim(s)
proof
  assume
A1: s is convergent;
  set g1=lim(s);
  set g=z*g1;
A2: 0+0< ||.z.||+1 by NORMSP_1:4,XREAL_1:8;
A3: 0<=||.z.|| by NORMSP_1:4;
A4: now
    let p be Real;
    assume 0<p;
    then
A5: 0<p/(||.z.||+1) by A2,XREAL_1:139;
    then consider n such that
A6: for m st n<=m holds ||.s.m-g1.||<p/(||.z.||+1) by A1,NORMSP_1:def 7;
    take n;
    let m;
    assume n<=m;
    then
A7: ||.s.m-g1.||< p/(||.z.||+1) by A6;
    0<=||.s.m-g1.|| by NORMSP_1:4;
    then
A8: ||.z.||*||.s.m-g1.||<= ||.z.||*(p/(||.z.||+1)) by A3,A7,XREAL_1:66;
    ||.z*(s.m-g1).|| <= ||.z.||*||.s.m-g1.|| by LOPBAN_3:38;
    then
A9: ||.z*(s.m-g1).||<=||.z.||*(p/(||.z.||+1)) by A8,XXREAL_0:2;
A10: ||.((z*s).m)-g.|| =||.z*s.m-z*g1.|| by LOPBAN_3:def 5
      .=||.z*(s.m-g1).|| by LOPBAN_3:38;
    0+ ||.z.|| < ||.z.||+1 by XREAL_1:8;
    then
A11: ||.z.||*(p/(||.z.||+1)) < ( ||.z.||+1) *(p/(||.z.||+1)) by A3,A5,
XREAL_1:97;
    ( ||.z.||+1) *(p/(||.z.||+1)) =p by A2,XCMPLX_1:87;
    hence ||.((z*s).m)-g.|| < p by A10,A9,A11,XXREAL_0:2;
  end;
  z*s is convergent by A1,Th4;
  hence thesis by A4,NORMSP_1:def 7;
end;
