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 Th3:
     n*0.L = 0.L
     proof
       defpred P[Nat] means $1*0.L = 0.L;
A1:    for n be Nat st P[n] holds P[n+1]
       proof
         let n be Nat;
         assume
A2:      P[n];
         (n+1)*0.L = n*0.L + 1*0.L by BINOM:15
         .= 0.L by A2,BINOM:13;
         hence thesis;
       end;
A3:    P[0] by BINOM:12;
       for n be Nat holds P[n] from NAT_1:sch 2(A3,A1);
       hence thesis;
     end;
