defpred S₁[ Nat] means x |^ x is integer ;
A1: S₁[ 0 ] by NEWTON:4;

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 A3: S₁[k] ; :: thesis: S₁[k + 1]
x |^ (k + 1) = (x |^ k) * x by NEWTON:6;
hence S₁[k + 1] by A3; :: thesis: verum

end;

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