defpred S₁[ Nat] means a |^ R <> 0. R;
a |^ 0 = 1_ R by BINOM:8;
then IA: S₁[ 0 ] ;

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]
a |^ (k + 1) = (a |^ k) * (a |^ 1) by BINOM:10
.= (a |^ k) * a by BINOM:8 ;
hence S₁[k + 1] by IV, VECTSP_2:def 1; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
hence not a |^ n is zero ; :: thesis: verum