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 Th5:
  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;
  let p;
  assume
A5: 0<p;
  then consider n1 such that
A6: for m st n1<=m holds |.seq.m-g1.|<p/2 by A3;
  consider n2 such that
A7: for m st n2<=m holds |.seq9.m-g2.|<p/2 by A4,A5;
  take k=n1+n2;
  let m such that
A8: k<=m;
  n1<=n1+n2 by NAT_1:12;
  then n1<=m by A8,XXREAL_0:2;
  then
A9: |.seq.m-g1.|<p/2 by A6;
  n2<=k by NAT_1:12;
  then n2<=m by A8,XXREAL_0:2;
  then |.seq9.m-g2.|<p/2 by A7;
  then
A10: |.seq.m-g1.|+|.seq9.m-g2.|<p/2+p/2 by A9,XREAL_1:8;
A11: |.(seq+seq9).m-g.|=|.seq.m+seq9.m-(g1+g2).| by SEQ_1:7
    .=|.seq.m-g1+(seq9.m-g2).|;
  |.seq.m-g1+(seq9.m-g2).|<=|.seq.m-g1.|+|.seq9.m-g2.| by COMPLEX1:56;
  hence thesis by A10,A11,XXREAL_0:2;
end;
