
theorem Th1:
  for c be Real, seq be Real_Sequence st seq is convergent for rseq
  be Real_Sequence st ( for i be Nat holds rseq .i = |.seq.i-c.| )
  holds rseq is convergent & lim rseq = |.lim seq-c.|
proof
  let c be Real;
  reconsider cc=c as Element of REAL by XREAL_0:def 1;
  set cseq = seq_const c;
  let seq be Real_Sequence such that
A1: seq is convergent;
A2: seq -cseq is convergent by A1;
  then
A3: abs(seq -cseq) is convergent;
  let rseq be Real_Sequence such that
A4: for i be Nat holds rseq .i = |.seq.i-c.|;
  now
    let i be Nat;
    thus rseq.i=|.seq.i-c.| by A4
      .=|.seq.i-(cseq.i).|by SEQ_1:57
      .=|.seq.i+-(cseq.i).|
      .=|.seq.i+(-cseq).i.| by SEQ_1:10
      .=|.(seq+(-cseq)).i.| by SEQ_1:7
      .=|.(seq -cseq ).i.| by SEQ_1:11
      .=abs(seq -cseq).i by SEQ_1:12;
  end;
  then
A5: for x be object st x in NAT holds rseq.x = abs(seq -cseq).x;
  lim abs(seq -cseq) = |.lim(seq-cseq).| by A2,SEQ_4:14
    .=|.lim seq-lim cseq.| by A1,SEQ_2:12
    .=|.lim seq-(cseq.0).| by SEQ_4:26
    .=|.lim seq-c.| by SEQ_1:57;
  hence thesis by A5,A3,FUNCT_2:12;
end;
