
theorem Th1:
  for c be Complex, seq be Complex_Sequence, rseq be Real_Sequence
st seq is convergent & ( for i be Nat holds rseq.i = |.(seq.i-c).| )
  holds rseq is convergent & lim rseq = |.(lim seq-c).|
proof
  let c be Complex;
  let seq be Complex_Sequence;
  let rseq be Real_Sequence;
  assume that
A1: seq is convergent and
A2: for i be Nat holds rseq.i = |.(seq.i-c).|;
  reconsider c1 = c as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider cseq = NAT --> c1 as Complex_Sequence;
A3: for n be Nat holds cseq.n=c
    by ORDINAL1:def 12,FUNCOP_1:7;
  then
A4: cseq is convergent by COMSEQ_2:9;
  then reconsider seq9 = seq-cseq as convergent Complex_Sequence by A1;
  seq -cseq is convergent by A1,A4;
  then
A5: lim |.(seq -cseq).| = |.(lim(seq-cseq)).| by SEQ_2:27
    .=|.(lim seq-lim cseq).| by A1,A4,COMSEQ_2:26
    .=|.(lim seq-c).| by A3,COMSEQ_2:10;
  now
    let i be Nat;
A6:  i in NAT by ORDINAL1:def 12;
    thus rseq.i=|.(seq.i-c).| by A2
      .=|.(seq.i-(cseq.i)).|by FUNCOP_1:7,A6
      .=|.(seq.i+-(cseq.i)).|
      .=|.(seq.i+(-cseq).i).| by VALUED_1:8
      .=|.(seq -cseq).i .| by VALUED_1:1,A6
      .=|. seq-cseq .| .i by VALUED_1:18;
  end;
  then
A7: for x be object st x in NAT holds rseq.x = |.(seq -cseq).| .x;
  |.seq9.| is convergent;
  hence thesis by A7,A5,FUNCT_2:12;
end;
