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 Th8:
  seq is convergent implies lim(r(#)seq)=r*(lim seq)
proof
  assume
A1: seq is convergent;
A2: now
    assume
A3: r=0;
    let p such that
A4: 0<p;
    reconsider k=0 as Nat;
    take k;
    let m such that k<=m;
    |.((r(#)seq).m)-r*(lim seq).|=|.0*seq.m-0.| by A3,SEQ_1:9
      .=0 by ABSVALUE:2;
    hence |.((r(#)seq).m)-r*(lim seq).|<p by A4;
  end;
  now
    assume
A5: r<>0;
    then
A6: 0<|.r.| by COMPLEX1:47;
    let p such that
A7: 0<p;
A8: 0<>|.r.| by A5,COMPLEX1:47;
    0<p/|.r.| by A6,A7;
    then consider n1 such that
A9: for m st n1<=m holds |.seq.m-(lim seq).|<p/|.r.| by A1,Def6;
    take k=n1;
    let m;
    assume k<=m;
    then
A10: |.seq.m-(lim seq).|<p/|.r.| by A9;
A11: |.((r(#)seq).m)-r*(lim seq).|=|.r*seq.m-r*(lim seq).| by SEQ_1:9
      .=|.r*(seq.m-(lim seq)).|
      .=|.r.|*|.seq.m-(lim seq).| by COMPLEX1:65;
    |.r.|*(p/|.r.|)=|.r.|*((|.r.|)"*p) by XCMPLX_0:def 9
      .=|.r.|*(|.r.|)"*p
      .=1*p by A8,XCMPLX_0:def 7
      .=p;
    hence |.((r(#)seq).m)-r*(lim seq).|<p by A6,A10,A11,XREAL_1:68;
  end;
  hence thesis by A1,A2,Def6;
end;
