defpred S₁[ Nat] means $1 * (0. R) = 0. R;
IA: S₁[ 0 ] by BINOM:12;

IS: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

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

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
hence n * (0. R) = 0. R ; :: thesis: verum