
theorem FIC:
  for seq be Complex_Sequence, x be Nat holds
  (for k be Nat st k >= x holds seq.k = 0) implies seq is summable
  proof
    let seq be Complex_Sequence, x be Nat;
    assume
    A1: for k be Nat holds (k >= x implies seq.k = 0);
    for k be Nat holds (seq^\x).k = 0
    proof
      let k be Nat;
      B1: k + x >= 0 + x by XREAL_1:6;
      (seq^\x).k = seq.(k+x) by NAT_1:def 3;
      hence thesis by A1,B1;
    end; then
    seq^\x is empty-yielding;
    hence thesis by COMSEQ_3:59;
  end;
