reserve x,y,z for object;
reserve n,m,k for Element of NAT;
reserve r for Real;

theorem
  for n being Nat ex m st n = 2*m or n = 2*m+1
proof
  let n be Nat;
  take n div 2;
  set k = n mod 2;
A1: k = 0 or k = 1
  proof
    k < 1 + 1 by NAT_D:1;
    then
A2: k <= 0 + 1 by NAT_1:13;
    now
      per cases by A2,NAT_1:8;
      suppose
        k <= 0;
        hence thesis by NAT_1:2;
      end;
      suppose
        k = 0 + 1;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  n = 2*(n div 2) + (n mod 2) by NAT_D:2;
  hence thesis by A1;
end;
