defpred S₁[ Nat] means p `^ R <> 0_. R;
p `^ 0 = 1_. R by POLYNOM5:15;
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]

now :: thesis: not p `^ (k + 1) = 0_. R

assume A: p `^ (k + 1) = 0_. R ; :: thesis: contradiction
p `^ (k + 1) = (p `^ k) *' p by POLYNOM5:19;
hence contradiction by IV, A, UPROOTS:21; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
hence not p `^ n is zero by UPROOTS:def 5; :: thesis: verum