let i, k be Nat; :: thesis: ( not i is even implies not i |^ k is even )
assume A1: not i is even ; :: thesis: not i |^ k is even
A2: i |^ 0 = 1 by NEWTON:9;
defpred S₁[ Nat] means not i |^ $1 is even ;
1 mod 2 = 1 by NAT_D:24;
then A3: S₁[ 0 ] by A2, NAT_2:24;
A4: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A5: not i |^ n is even ; :: thesis: S₁[n + 1]
i |^ (n + 1) = (i |^ n) * i by NEWTON:11;
hence S₁[n + 1] by A1, A5; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A4);
hence not i |^ k is even ; :: thesis: verum