let L be non empty right_complementable associative Abelian add-associative right_zeroed distributive left_zeroed doubleLoopStr ; :: thesis: for n being Nat holds n * (0. L) = 0. L
let n be Nat; :: thesis: n * (0. L) = 0. L
defpred S₁[ Nat] means $1 * (0. L) = 0. L;
A1: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A2: S₁[n] ; :: thesis: S₁[n + 1]
(n + 1) * (0. L) = (n * (0. L)) + (1 * (0. L)) by BINOM:15
.= 0. L by A2, BINOM:13 ;
hence S₁[n + 1] ; :: thesis: verum

end;

A3: S₁[ 0 ] by BINOM:12;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A1);
hence n * (0. L) = 0. L ; :: thesis: verum