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

proof

let m be natural number ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume a |^ m <> 0 ; :: thesis: S₁[m + 1]
then (a |^ m) * a <> 0 by A1, XCMPLX_1:6;
hence S₁[m + 1] by NEWTON:6; :: thesis: verum

end;

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