 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Lm11:
  for n,m be Nat,b be Element of R holds
    (n*m)*b = n*(m*b)
   proof
     let n,m be Nat,b be Element of R;
     defpred P[Nat] means ($1*m)*b = $1*(m*b);
A1:  for n be Nat st P[n] holds P[n+1]
     proof
       let n be Nat;
       assume
A2:    P[n];
       ((n+1)*m)*b = ((n*m) + (1*m))*b
         .= (n*m)*b + m*b by BINOM:15
         .= n*(m*b) + 1*(m*b) by A2,BINOM:13
         .= (n+1)*(m*b) by BINOM:15;
       hence thesis;
     end;
     (0*m)*b = 0.R by BINOM:12
       .= 0*(m*b) by BINOM:12; then
A3:  P[0];
     for n be Nat holds P[n] from NAT_1:sch 2(A3,A1);
     hence thesis;
   end;
