Before we can even begin to discuss whatever you are trying to prove, I must make the following observations:
1. Commercial banks do not primarily rely on deposit money, but rather on reserve money created by the central bank against government debt. So the money commercial banks loan is always debt, issued either by the central bank or themselves.
2. It is impossible for the public to keep 90% of commercial bank money "under the mattress" because:
2.1. Less than 3% of the money supply exist in physical form.
2.2. Loans must be repaid with interest, so the public must not only return that money to their creditor banks but also give them additional money, which in turn must be created by the central bank against additional government debt.
I think most of these are lies that told by reserve banks. Due to the complex nature of "LOAN", maybe only one in a thousand understand how it really works. But looking from the view of the central bank, things are much eaiser to understand
A loan is a very complex operation, it involves several steps:
1. Banks first acquire the base money. Without base money, they can't loan out anything. They can get base money from their own savings, from FED or other financial institutions, when they are running out of base money, they will broke like Lehman
2. Once they had some base money, say $100, they could loan out $90 to the borrower, they credit the borrower's account with $90, at the same time, they must deposite $10 to FED as reserve
3. Since the borrower's account are credited with those $90, the effect is that the money they loaned out are immediately deposited back into the banking system (either in the same bank or another bank), thus the base money never leave the banking system. This happened almost instantly. Now they have the base money back, they could lend those out again, only 10% less each time, due to reserve requirement
But this chicken and egg loop require that they must have base money at the first place. Historically base money is gold, produced by miners, but now it is created by FED, produced by adding numbers in the checkbook
Money and debt are totally seperate concept, banks always trying to combine them together just to disguise their scam. Money is just a measure of the value of the debt, you can pay back the debt using anything like labor, security, assets, etc...