
theorem Th17:
  for f be Real_Sequence,k,n be Nat st k in Seg n holds (f |_ Seg n).k = f.k
  proof
    let f be Real_Sequence, k,n be Nat;
    assume
A0: k in Seg n;
A1: dom f = NAT by FUNCT_2:def 1;
    dom (f | Seg n) = Seg n by A1,RELAT_1:62; then
    (f |_ Seg n).k = (f | Seg n).k by A0,FUNCT_4:13  .= f.k by A0,FUNCT_1:49;
    hence thesis;
  end;
