reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th11:
  s is convergent implies lim(s*') = (lim s)*'
proof
  set g = lim s;
  assume
A1: s is convergent;
  then reconsider s1 = s as convergent Complex_Sequence;
A2: now
    let p be Real;
    assume 0<p;
    then consider n such that
A3: for m st n<=m holds |.s.m-g.|<p by A1,Def6;
    take n;
    let m such that
A4: n<=m;
    m in NAT by ORDINAL1:def 12;
    then |.(s*').m - g*'.| = |.(s.m)*' - g*'.| by Def2
      .= |.(s.m - g)*'.| by COMPLEX1:34
      .= |.s.m - g.| by COMPLEX1:53;
    hence |.(s*').m - (lim s)*'.| < p by A3,A4;
  end;
  s1*' is convergent;
  hence thesis by A2,Def6;
end;
