defpred S₁[ Nat] means ( not m is empty implies 2 |^ m is even );
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume that
S₁[k] and
not k + 1 is empty ; :: thesis: 2 |^ (k + 1) is even
2 |^ (k + 1) = 2 * (2 |^ k) by NEWTON:11;
hence 2 |^ (k + 1) is even ; :: thesis: verum

end;

A2: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A1);
hence 2 |^ m is even ; :: thesis: verum