WOW, I apologize for all the answers you have received, the clearly don't understand your question or delayed financing.
Your answer is yes.
If you cash out on your current mortgage to purchase an investment property with all cash, you may use the delayed financing to secure a mortgage on the investment property , however you will be required to use these funds to pay down your mortgage that you cashed out.
From Fannie Mae:
"If the source of funds used to acquire the property was an unsecured loan or a loan secured by an asset other than the subject property (such as a HELOC secured by another property), the HUD-1 for the refinance transaction must reflect that all cash- out proceeds be used to pay down, if applicable, the loan (unsecured or secured by an asset other than the subject property) used to purchase the property. Any payments on the balance remaining from the original loan must be included in the debt-to-income ratio calculation for the refinance transaction."
Please contact me if you need more information.