 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;
 reserve g for non zero Polynomial of INT.Ring;

theorem Th15:
  for f be Element of the carrier of Polynom-Ring INT.Ring holds
  for i be Nat holds
  ((tau(0)) *'f).(i+1) = f.i & ((tau(0)) *'f).0 = 0.INT.Ring
   proof
     let f be Element of the carrier of Polynom-Ring INT.Ring;
     for i be Nat holds ((tau(0)) *'f).(i+1) = f.i
     & ((tau(0)) *'f).0 = 0.INT.Ring
     proof
       let i be Nat;
A1:    i in NAT by ORDINAL1:def 12;
       tau(0) = <%0.INT.Ring,1.INT.Ring%>`^1 by POLYNOM5:16
       .= anpoly(1.INT.Ring,1) by FIELD_1:12;
       hence thesis by A1,RINGDER1:31;
     end;
     hence thesis;
   end;
