In the above,

Consider the string y' such that the length of the resulting stack content is minimum i.e.

          for some , then

So, once P reaches the configuration after processing xy', it cannot remove any symbol from the stock (since length cannot be reduced further) in the subsequent moves. Because a move of a DPDA in standard from that involves removing a single symbol from the stack reduces the height of the stack.

Let for some and . Since we may consider any string in xy' there are infinite number of strings of the form xy . But the set of states and stack symbols, , respectively of P are infinite. Hence there must exist two different strings u=xy' and v=xy' in . Such that

                  
    and