defpred S₁[ Nat] means (0. R) |^ $1 = 0. R;
IA: S₁[1] by BINOM:8;

IS: 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 IV: S₁[k] ; :: thesis: S₁[k + 1]
(0. R) |^ (k + 1) = (0. R) * ((0. R) |^ 1) by IV, BINOM:10;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being non zero Nat holds S₁[k] from NAT_1:sch 10(IA, IS);
hence (0. R) |^ k = 0. R ; :: thesis: verum