How should the ownership and profits from the sale of a property be divided between two partners if the initial down payment came from one partner, and the subsequent financing relied on the other partner's creditworthiness, with both of them paying the installments equally?

It is permissible for two people to jointly purchase a property, with each owning a percentage of it according to their contribution to the paid amount, and profit and loss following the same proportion. The method of financing or the source of the down payment does not alter this rule. If the bank purchases the property and then sells it to both of you in installments, this is a permissible Murabaha sale, subject to its conditions. Paying a earnest deposit during the binding promise stage (for Murabaha or Ijara wa Iqtina) has been permitted by some scholars on the condition that the bank does not dispose of it, but the fatwa on this site is that it is not permissible. If the bank leases the property to both of you through Ijara wa Iqtina (lease ending in ownership), it is permissible, subject to its conditions, and you would be partners in the property according to the percentage each of you paid.