reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th21:
  k > 0 & i is even implies i |^ k is even
proof
  assume that
A1: k > 0 and
A2: i is even;
  defpred P[Nat] means $1 > 0 & i is even implies i |^ $1 is even;
A3: for n holds P[n] implies P[n+1]
  proof
    let n;
    assume P[n];
    P[n+1]
    proof
      now
          (i |^ n)*i is even by A2;
          hence thesis by NEWTON:6;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
A4: P[0];
  for n holds P[n] from NAT_1:sch 2(A4,A3);
  hence thesis by A1,A2;
end;
