defpred S₁[ Nat] means n |^|^ n is natural ;
A0: S₁[ 0 ] by Th0;

A1: now

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then reconsider nk = n |^|^ k as natural number ;
k + 1 = succ k by NAT_1:39;
then n |^|^ (k + 1) = exp n,nk by Th1;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A0, A1);
hence n |^|^ k is natural ; :: thesis: verum