
theorem Th40:
  for L being non empty ZeroStr, n being Element of NAT, p being
  Polynomial of L st n >= len p holds poly_shift(p,n) = 0_. L
proof
  let L be non empty ZeroStr, n be Element of NAT, p be Polynomial of L;
  set ps = poly_shift(p,n);
  assume
A1: n >= len p;
A2: now
    let z be object;
    assume z in dom ps;
    then reconsider i = z as Element of NAT;
    thus ps.z = p.(n+i) by Def5
      .= 0.L by A1,ALGSEQ_1:8,NAT_1:12;
  end;
  dom ps = NAT by FUNCT_2:def 1;
  hence thesis by A2,FUNCOP_1:11;
end;
