defpred S₁[ Nat] means n |^|^ n is natural ;
A1: S₁[ 0 ] by Th13;

A2: 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 S₁[k] ; :: thesis: S₁[k + 1]
then reconsider nk = n |^|^ k as Nat ;
Segm (k + 1) = succ (Segm k) by NAT_1:38;
then n |^|^ (k + 1) = exp (n,nk) by Th14;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence n |^|^ k is natural ; :: thesis: verum