defpred S₁[ Nat] means a |^ a is rational ;
A1: S₁[ 0 ] by NEWTON:4;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
a |^ (k + 1) = (a |^ k) * a by NEWTON:6;
hence ( S₁[k] implies S₁[k + 1] ) ; :: thesis: verum

end;

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