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

theorem Th11:
  for p be odd prime Nat, m be positive Nat holds
  len ~(Product (x.(m,p))) = m*p + 1
    proof
      let p be odd prime Nat, m be positive Nat;
      set PR = Polynom-Ring INT.Ring;
      defpred P[non zero Nat] means len ~(Product (x.($1,p))) = $1*p +1;
A1:   P[1]
      proof
A2:     len x.(1,p) = 1 by Def2;
A3:     x.(1,p) = <* (x.(1,p)).1 *> by Def2,FINSEQ_1:40;
        dom x.(1,p) = Seg 1 by A2,FINSEQ_1:def 3; then
        1 in dom x.(1,p); then
        (x.(1,p)).1 = (tau(1))|^p by Def2; then
        len ~Product (x.(1,p)) = len ~((tau(1))|^p) by A3,FINSOP_1:11
        .= p + 1 by Th8;
        hence thesis;
      end;
A5:   for k be non zero Nat holds P[k] implies P[k+1]
      proof
        let k be non zero Nat;
        assume
A6:     P[k];
        set pp0 = Product(x.(k,p)), t0 = (tau(k+1))|^p;
A7:     len ~((tau(k+1))|^p) = p+1 by Th8; then
A8:     (len ~pp0)*(len ~t0) <> 0 by A6;
        len ~(Product (x.(k+1,p)))
        = len ~(pp0*t0) by Th10
        .= (len ~pp0) + (len ~t0) -1 by A8,Th9
        .= (k+1)*p + 1 by A6, A7;
        hence thesis;
      end;
      for k be non zero Nat holds P[k] from NAT_1:sch 10(A1,A5);
      hence thesis;
    end;
