 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;

theorem Th6:  :: **
  for n be Nat,
      f be Element of the carrier of Polynom-Ring INT.Ring
   holds n*f = In(n,INT.Ring)*f
   proof
     set R = INT.Ring, PR = Polynom-Ring INT.Ring;
     let n be Nat,
     f be Element of the carrier of Polynom-Ring INT.Ring;
     defpred P[Nat] means $1*f = In($1,INT.Ring)*f;
     In(0,INT.Ring)*f = 0_.R by POLYNOM5:26
       .= 0.PR by POLYNOM3:def 10; then
A1:  P[0] by BINOM:12;
A2:  for k be Nat holds P[k] implies P[k+1]
     proof
       let k be Nat;
       assume
A3:    P[k];
       set r = In(k,INT.Ring);
A4:    In(k,INT.Ring)*f + f = r*f + 1.R*f by POLYALG1:9
       .= (r+1.R)*f by POLYALG1:7
       .= In(k+1,INT.Ring)*f;
       In(k,INT.Ring)*f + f = k*f + f by A3,POLYNOM3:def 10
       .= k*f + 1*f by BINOM:13 .= (k+1)*f by BINOM:15;
       hence thesis by A4;
     end;
     for k be Nat holds P[k] from NAT_1:sch 2(A1,A2);
     hence thesis;
   end;
