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 Th6:
  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;
  now
    let p;
    assume 0<p;
    then
A3: 0<p/2;
    then consider n1 such that
A4: for m st n1<=m holds |.seq.m-(lim seq).|<p/2 by A1,Def6;
    consider n2 such that
A5: for m st n2<=m holds |.seq9.m-(lim seq9).|<p/2 by A2,A3,Def6;
    take k=n1+n2;
    let m such that
A6: k<=m;
    n1<=n1+n2 by NAT_1:12;
    then n1<=m by A6,XXREAL_0:2;
    then
A7: |.seq.m-(lim seq).|<p/2 by A4;
    n2<=k by NAT_1:12;
    then n2<=m by A6,XXREAL_0:2;
    then |.seq9.m-(lim seq9).|<p/2 by A5;
    then
A8: |.seq.m-(lim seq).|+|.seq9.m-(lim seq9).|<p/2+p/2 by A7,XREAL_1:8;
A9: |.((seq+seq9).m)-((lim seq)+(lim seq9)).|
    =|.seq.m+seq9.m-((lim seq)+(lim seq9)).| by SEQ_1:7
      .=|.seq.m-(lim seq)+(seq9.m-(lim seq9)).|;
    |.seq.m-(lim seq)+(seq9.m-(lim seq9)).|<=
    |.seq.m-(lim seq).|+|.seq9.m-(lim seq9).| by COMPLEX1:56;
    hence |.(seq+seq9).m-((lim seq)+(lim seq9)).|<p by A8,A9,XXREAL_0:2;
  end;
  hence thesis by A1,A2,Def6;
end;
