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 Th12:
 for s,s9 being convergent Complex_Sequence
  holds lim (s + s9)=(lim s)+ (lim s9)
proof
 let s,s9 be convergent Complex_Sequence;
  for p be Real st 0<p ex n st for m st n<=m
   holds |.(s + s9).m-((lim s)+(
  lim s9)).|<p
  proof
    let p be Real;
    assume 0<p;
    then
A1: 0<p/2;
    then consider n1 such that
A2: for m st n1<=m holds |.s.m-lim s.|<p/2 by Def6;
    consider n2 such that
A3: for m st n2<=m holds |.s9.m-lim s9.|<p/2 by A1,Def6;
    reconsider n = max(n1,n2) as Element of NAT by ORDINAL1:def 12;
    take n;
    let m such that
A4: n<=m;
    n2<=n by XXREAL_0:25;
    then n + n2 <= n + m by A4,XREAL_1:7;
    then n2<=m by XREAL_1:6;
    then
A5: |.s9.m-lim s9.|<p/2 by A3;
    m in NAT by ORDINAL1:def 12;
    then
A6: |.(s + s9).m-((lim s)+(lim s9)).| = |.s.m + s9.m - ((lim s)+(lim s9))
    .| by VALUED_1:1
      .=|.(s.m -lim s) + (s9.m - lim s9).|;
    n1<=n by XXREAL_0:25;
    then n + n1 <= n + m by A4,XREAL_1:7;
    then n1<=m by XREAL_1:6;
    then |.s.m-lim s.|<p/2 by A2;
    then |.s.m-lim s.| + |.s9.m-lim s9.| < p/2 + p/2 by A5,XREAL_1:8;
    then |.s.m-lim s.| + |.s9.m -lim s9.| + |.(s + s9).m-((lim s)+(lim s9)).|
    < p + (|.s.m-lim s.| + |.s9.m -lim s9.|) by A6,COMPLEX1:56,XREAL_1:8;
    hence thesis by XREAL_1:6;
  end;
  hence thesis by Def6;
end;
