
theorem RSH:
  for f be complex-valued Sequence, n be Nat holds
  Re Shift (f,n) = Shift (Re f,n) & Im Shift (f,n) = Shift (Im f,n)
  proof
    let f be complex-valued Sequence, n be Nat;
    A0: dom Re (Shift (f,n)) = dom Shift (f,n) & dom Re f = dom f
      by COMSEQ_3:def 3; then
    A1: dom Re(Shift (f,n)) = {m+n where m is Nat: m in dom f}
      by VALUED_1:def 12
    .= dom (Shift (Re f,n)) by A0,VALUED_1:def 12;
    A0a: dom Im Shift (f,n) = dom Shift (f,n) & dom Im f = dom f
      by COMSEQ_3:def 4; then
    A2: dom Im Shift (f,n) = {m+n where m is Nat: m in dom f}
      by VALUED_1:def 12
    .= dom (Shift (Im f,n)) by A0a,VALUED_1:def 12;
    A3: for x be object st x in dom Re Shift (f,n) holds
      (Re (Shift (f,n))).x = (Shift (Re f,n)).x
    proof
      let x be object;
      assume B1: x in dom (Re(Shift (f,n))); then
      consider k be Nat such that
      B2: k in dom f & x = k + n by A0,VALUED_1:39;
      (Re (Shift (f,n))).x = Re ((Shift (f,n)).(k+n)) by B1,B2,COMSEQ_3:def 3
      .= Re (f.k) by B2,VALUED_1:def 12
      .= (Re f).k by A0,B2,COMSEQ_3:def 3
      .= (Shift ((Re f),n)).(k+n) by A0,B2,VALUED_1:def 12;
      hence thesis by B2;
    end;
    for x be object st x in dom (Im(Shift (f,n))) holds
      (Im (Shift (f,n))).x = (Shift ((Im f),n)).x
    proof
      let x be object such that
      B1: x in dom Im Shift (f,n);
      consider k be Nat such that
      B2: k in dom f & x = k + n by B1,A0a,VALUED_1:39;
      (Im (Shift (f,n))).x = Im ((Shift (f,n)).(k+n)) by B1,B2,COMSEQ_3:def 4
      .= Im (f.k) by B2,VALUED_1:def 12
      .= (Im f).k by A0a,B2,COMSEQ_3:def 4
      .= (Shift (Im f,n)).(k+n) by A0a,B2,VALUED_1:def 12;
      hence thesis by B2;
    end;
    hence thesis by A1,A2,A3;
  end;
