reserve L for Abelian left_zeroed add-associative associative right_zeroed
              right_complementable distributive non empty doubleLoopStr;
reserve a,b,c for Element of L;
reserve R for non degenerated comRing;
reserve n,m,i,j,k for Nat;

theorem
     0.L*n = 0.L
     proof
       defpred P[Nat] means 0.L * $1= 0.L;
A1:    for n be Nat st P[n] holds P[n+1]
       proof
         let n be Nat;
         assume P[n];
         0.L*(n+1) = (n+1)*0.L by BINOM:17
         .= n*0.L + 1*0.L by BINOM:15
         .= n*0.L + 0.L by BINOM:13
         .= 0.L by Th3;
         hence thesis;
       end;
A2:    P[0] by BINOM:12;
       for n be Nat holds P[n] from NAT_1:sch 2(A2,A1);
       hence thesis;
     end;
