There are amultitude of logical systems. Each are self-consistent, meaning that a number of theorems can be proven in each logical system without encountering contradictions. In such systems it is possible to replace an assumption by a theorem and the entire logical system of theorems and the missing postulate can be derived. Proceding assumption by assumption or postulate by postulate all of the original ones can be replaced by what were originally theorems.
It`s too tedious to do this for particular systems in this forum. One thing missing is that there are some correct theorems in almost any sufficiently complex system that cannot be derived from the postulates or even the theorems acting as postulates. Godel proved this. That this is true, means that not only are logical systems merely circular, they are not even rigorous. Like Swiss cheese, there are holes in every complex logical system.