let i, k be Nat; :: thesis: ( i is odd implies i |^ k is odd )
defpred S₁[ Nat] means i |^ $1 is odd ;
assume A1: i is odd ; :: thesis: i |^ k is odd
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
A3: i |^ (n + 1) = (i |^ n) * i by NEWTON:6;
assume i |^ n is odd ; :: thesis: S₁[n + 1]
hence S₁[n + 1] by A1, A3; :: thesis: verum

end;

( i |^ 0 = 1 & 1 mod 2 = 1 ) by NAT_D:24, NEWTON:4;
then A4: S₁[ 0 ] by NAT_2:22;
for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A2);
hence i |^ k is odd ; :: thesis: verum