let a be Real; :: thesis: for n being Nat st 0 < a holds
0 < a |^ n
let n be Nat; :: thesis: ( 0 < a implies 0 < a |^ n )
defpred S₁[ Nat] means 0 < a |^ $1;
assume A1: 0 < a ; :: thesis: 0 < a |^ n
A2: for m being Nat st S₁[m] holds
S₁[m + 1]

proof

let m be Nat; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume a |^ m > 0 ; :: thesis: S₁[m + 1]
then (a |^ m) * a > 0 * a by A1, XREAL_1:68;
hence S₁[m + 1] by NEWTON:6; :: thesis: verum

end;

A3: S₁[ 0 ] by NEWTON:4;
for m being Nat holds S₁[m] from NAT_1:sch 2(A3, A2);
hence 0 < a |^ n ; :: thesis: verum