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 Th7:
  seq is convergent implies r(#)seq is convergent
proof
  assume seq is convergent;
  then consider g1 such that
A1: for p st 0<p ex n st for m st n<=m holds |.seq.m-g1.|<p;
  take g=r*g1;
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)-g.|=|.0*seq.m-0.| by A3,SEQ_1:9
      .=0 by ABSVALUE:2;
    hence |.((r(#)seq).m)-g.|<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;
    consider n1 such that
A9: for m st n1<=m holds |.seq.m-g1.|<p/|.r.| by A1,A6,A7;
    take k=n1;
    let m;
    assume k<=m;
    then
A10: |.seq.m-g1.|<p/|.r.| by A9;
A11: |.((r(#)seq).m)-g.|=|.r*seq.m-r*g1.| by SEQ_1:9
      .=|.r*(seq.m-g1).|
      .=|.r.|*|.seq.m-g1.| 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)-g.|<p by A6,A10,A11,XREAL_1:68;
  end;
  hence thesis by A2;
end;
