let R be Ring; :: thesis: for n being non zero Nat holds (power R) . ((0. R),n) = 0. R
let n be non zero Nat; :: thesis: (power R) . ((0. R),n) = 0. R
defpred S₁[ Nat] means (power R) . ((0. R),$1) = 0. R;
(power R) . ((0. R),(0 + 1)) = ((power R) . ((0. R),0)) * (0. R) by GROUP_1:def 7
.= 0. R ;
then A1: S₁[1] ;

A2: now :: thesis: for k being non zero Nat st S₁[k] holds
S₁[k + 1]

let k be non zero Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
(power R) . ((0. R),(k + 1)) = ((power R) . ((0. R),k)) * (0. R) by GROUP_1:def 7
.= 0. R ;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being non zero Nat holds S₁[k] from NAT_1:sch 10(A1, A2);
hence (power R) . ((0. R),n) = 0. R ; :: thesis: verum