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 Th20:
 for s,s9 being convergent Complex_Sequence
  holds lim(s(#)s9)=(lim s)*(lim s9)
proof
  let s,s9 be convergent Complex_Sequence;
  consider R such that
A1: 0<R and
A2: for n holds |.s.n.|<R by Th8;
A3: 0+0<|.(lim s9).|+R by A1,COMPLEX1:46,XREAL_1:8;
A4: 0<=|.(lim s9).| by COMPLEX1:46;
  now
    let p be Real;
    assume 0<p;
    then
A5: 0<p/(|.(lim s9).|+R) by A3;
    then consider n1 such that
A6: for m st n1<=m holds |.s.m-(lim s).|<p/(|.(lim s9).|+R) by Def6;
    consider n2 such that
A7: for m st n2<=m holds |.s9.m-(lim s9).|<p/(|.(lim s9).|+R) by A5,Def6;
    take n=n1+n2;
    let m such that
A8: n<=m;
    n1<=n1+n2 by NAT_1:12;
    then n1<=m by A8,XXREAL_0:2;
    then
A9: |.s.m-(lim s).|<=p/(|.(lim s9).|+R) by A6;
    |.(s.m-(lim s))*(lim s9).| =|.(lim s9).|*|.s.m-(lim s).| by COMPLEX1:65;
    then
A10: |.(s.m-(lim s))*(lim s9).|<=|.(lim s9).|* (p/(|.(lim s9).|+R)) by A4,A9,
XREAL_1:64;
A11: 0<=|.s.m.| & 0<=|.s9.m-(lim s9).| by COMPLEX1:46;
    n2<=n by NAT_1:12;
    then n2<=m by A8,XXREAL_0:2;
    then
A12: |.s9.m-(lim s9).|<p/(|.(lim s9).|+R) by A7;
    |.((s(#)s9).m)-(lim s)*(lim s9).| =|.(s.m*s9.m-s.m*(lim s9)+s.m*(lim
    s9))- (lim s)*(lim s9).| by VALUED_1:5
      .=|.s.m*(s9.m-(lim s9))+ (s.m-(lim s))*(lim s9).|;
    then
A13: |.((s(#)s9).m)-(lim s)*(lim s9).|<= |.s.m*(s9.m-(lim s9)).|+|.(s.m-(
    lim s))*(lim s9).| by COMPLEX1:56;
    |.s.m.|<R by A2;
    then |.s.m.|*|.s9.m-(lim s9).|<R*(p/(|.(lim s9).|+R)) by A11,A12,XREAL_1:96
;
    then
A14: |.s.m*(s9.m-(lim s9)).|<R*(p/(|.(lim s9).|+R)) by COMPLEX1:65;
    R*(p/(|.(lim s9).|+R))+|.(lim s9).|*(p/(|.(lim s9).|+R)) =(p/(|.(lim
    s9).|+R))*(|.(lim s9).|+R)
      .=p by A3,XCMPLX_1:87;
    then |.s.m*(s9.m-(lim s9)).|+|.(s.m-(lim s))* (lim s9).|<p by A14,A10,
XREAL_1:8;
    hence |.((s(#)s9).m)-(lim s)*(lim s9).|<p by A13,XXREAL_0:2;
  end;
  hence thesis by Def6;
end;
