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 Th14:
  seq is convergent & seq9 is convergent implies seq (#) seq9 is convergent
proof
  assume that
A1: seq is convergent and
A2: seq9 is convergent;
  consider g1 such that
A3: for p st 0<p ex n st for m st n<=m holds |.seq.m-g1.|<p by A1;
  consider g2 such that
A4: for p st 0<p ex n st for m st n<=m holds |.seq9.m-g2.|<p by A2;
  take g=g1*g2;
  consider r such that
A5: 0<r and
A6: for n holds |.seq.n.|<r by A1,Th3;
A7: 0<=|.g2.| by COMPLEX1:46;
A8: 0+0<|.g2.|+r by A5,COMPLEX1:46,XREAL_1:8;
  let p;
  assume
A9: 0<p;
  then consider n1 such that
A10: for m st n1<=m holds |.seq.m-g1.|<p/(|.g2.|+r) by A3,A8;
  consider n2 such that
A11: for m st n2<=m holds |.seq9.m-g2.|<p/(|.g2.|+r) by A4,A8,A9;
  take n=n1+n2;
  let m such that
A12: n<=m;
A13: 0<=|.seq.m.| by COMPLEX1:46;
A14: 0<=|.seq9.m-g2.| by COMPLEX1:46;
  n2<=n by NAT_1:12;
  then n2<=m by A12,XXREAL_0:2;
  then
A15: |.seq9.m-g2.|<p/(|.g2.|+r) by A11;
  n1<=n1+n2 by NAT_1:12;
  then n1<=m by A12,XXREAL_0:2;
  then
A16: |.seq.m-g1.|<=p/(|.g2.|+r) by A10;
  |.((seq(#)seq9).m)-g.|=|.seq.m*seq9.m-(seq.m*g2-seq.m*g2)-g1*g2.| by
SEQ_1:8
    .=|.seq.m*(seq9.m-g2)+(seq.m-g1)*g2.|;
  then
A17: |.((seq(#)seq9).m)-g.|<=
  |.seq.m*(seq9.m-g2).|+|.(seq.m-g1)*g2.| by COMPLEX1:56;
  |.seq.m.|<r by A6;
  then |.seq.m.|*|.seq9.m-g2.|<r*(p/(|.g2.|+r)) by A13,A14,A15,XREAL_1:96;
  then
A18: |.seq.m*(seq9.m-g2).|<r*(p/(|.g2.|+r)) by COMPLEX1:65;
  |.(seq.m-g1)*g2.|=|.g2.|*|.seq.m-g1.| by COMPLEX1:65;
  then
A19: |.(seq.m-g1)*g2.|<=|.g2.|*(p/(|.g2.|+r)) by A7,A16,XREAL_1:64;
  r*(p/(|.g2.|+r))+|.g2.|*(p/(|.g2.|+r)) =(p/(|.g2.|+r))*(|.g2.|+r)
    .=p by A8,XCMPLX_1:87;
  then |.seq.m*(seq9.m-g2).|+|.(seq.m-g1)*g2.|<p by A18,A19,XREAL_1:8;
  hence thesis by A17,XXREAL_0:2;
end;
