reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem Th15:
  seq is convergent & seq9 is convergent implies
  lim(seq(#)seq9)=(lim seq)*(lim seq9)
proof
  assume that
A1: seq is convergent and
A2: seq9 is convergent;
   consider r such that
A3: 0<r and
A4: for n holds |.seq.n.|<r by A1,Th3;
A5: 0<=|.lim seq9.| by COMPLEX1:46;
A6: 0+0<|.lim seq9.|+r by A3,COMPLEX1:46,XREAL_1:8;
  now
    let p;
    assume 0<p;
    then
A7: 0<p/(|.lim seq9.|+r) by A6;
    then consider n1 such that
A8: for m st n1<=m holds |.seq.m-(lim seq).|<p/(|.lim seq9.|+r) by A1,Def6;
    consider n2 such that
A9: for m st n2<=m holds |.seq9.m-(lim seq9).|<p/(|.lim seq9.|+r)
    by A2,A7,Def6;
    take n=n1+n2;
    let m such that
A10: n<=m;
A11: 0<=|.seq.m.| by COMPLEX1:46;
A12: 0<=|.seq9.m-(lim seq9).| by COMPLEX1:46;
    n2<=n by NAT_1:12;
    then n2<=m by A10,XXREAL_0:2;
    then
A13: |.seq9.m-(lim seq9).|<p/(|.lim seq9.|+r) by A9;
    n1<=n1+n2 by NAT_1:12;
    then n1<=m by A10,XXREAL_0:2;
    then
A14: |.seq.m-(lim seq).|<=p/(|.lim seq9.|+r) by A8;
    |.((seq(#)seq9).m)-(lim seq)*(lim seq9).|
    =|.seq.m*seq9.m-(seq.m*(lim seq9)-seq.m*(lim seq9))-
    (lim seq)*(lim seq9).| by SEQ_1:8
      .=|.seq.m*(seq9.m-(lim seq9))+
    (seq.m-(lim seq))*(lim seq9).|;
    then
A15: |.((seq(#)seq9).m)-(lim seq)*(lim seq9).|<=
    |.seq.m*(seq9.m-(lim seq9)).|+
    |.(seq.m-(lim seq))*(lim seq9).| by COMPLEX1:56;
    |.seq.m.|<r by A4;
    then |.seq.m.|*|.seq9.m-(lim seq9).|<r*(p/(|.lim seq9.|+r))
    by A11,A12,A13,XREAL_1:96;
    then
A16: |.seq.m*(seq9.m-(lim seq9)).|<r*(p/(|.lim seq9.|+r)) by COMPLEX1:65;
    |.(seq.m-(lim seq))*(lim seq9).|
    =|.lim seq9.|*|.seq.m-lim seq.| by COMPLEX1:65;
    then
A17: |.(seq.m-(lim seq))*(lim seq9).|<=|.lim seq9.|*
    (p/(|.lim seq9.|+r)) by A5,A14,XREAL_1:64;
    r*(p/(|.lim seq9.|+r))+|.lim seq9.|*(p/(|.lim seq9.|+r))
    =(p/(|.lim seq9.|+r))*(|.lim seq9.|+r)
      .=p by A6,XCMPLX_1:87;
    then |.seq.m*(seq9.m-(lim seq9)).|+|.(seq.m-(lim seq))*
    (lim seq9).|<p by A16,A17,XREAL_1:8;
    hence |.((seq(#)seq9).m)-(lim seq)*(lim seq9).|<p by A15,XXREAL_0:2;
  end;
  hence thesis by A1,A2,Def6;
end;
