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

theorem Th9:
  for f,g be Element of the carrier of Polynom-Ring INT.Ring
  st (len ~f)*(len ~g) <> 0 holds len ~(f*g) = len ~f + len ~g -1
    proof
      let f,g be Element of the carrier of Polynom-Ring INT.Ring;
      reconsider p = ~f, q = ~g as Polynomial of INT.Ring;
      assume (len ~f)*(len ~g) <> 0; then
A2:   len p > 0 & len q > 0;
      p.(len p -'1) <> 0.INT.Ring & q.(len q -'1) <> 0.INT.Ring
        by A2,UPROOTS:18; then
      p.(len p -'1) * q.(len q -'1) <> 0.INT.Ring; then
      len (p*'q) = len p + len q - 1 by POLYNOM4:10;
      hence thesis by POLYNOM3:def 10;
    end;
