# About the ProvenMath Project

ProvenMath is a mathematical project which states the foundations of mathematics and derives mathematical knowledge from these foundations. Every theorem on ProvenMath is proved within the pages of ProvenMath and can be traced down to the foundations. The content of ProvenMath is self-contained. If you have found a theorem which interests you then you can be certain that you can also find a complete proof of this theorem.

All theorems on ProvenMath are given in this form: If [all assumptions enumerated] then [conclusion].
Each theorem on ProvenMath has all of its assumptions explicitly stated in the If section. Sometimes an assumption is written as a link to a page which explains it in more detail. All theorems have complete proofs.

All definitions and all conventions of notation are uniform throughout the pages of ProvenMath. If a mathematical term is used somewhere on ProvenMath you can be certain that it is defined within the pages of ProvenMath and you will be able to find the place of definition by following appropriate links or by performing a Google search within the pages of Apronus.com.

The pages of ProvenMath can be properly viewed under any Web browser because they are written only with the basic set of characters found on all keyboards and in all fonts. ProvenMath pages can be printed out easily.

ProvenMath is done by two people:
Michal Stanislaw Wojcik - graduate student of mathematics at Wroclaw University of Technology,
Michal Ryszard Wojcik - graduate student of mathematics at Wroclaw University of Technology.

We have personally gone through the proofs of a large portion of the mathematical knowledge that was presented in the course of our mathematical studies. Since the first year we have been documenting our knowledge in the form of hand-written notes in hard-cover notebooks. We wrote strict proofs for every theorem that we were learning. In the proofs we used only the theorems which we have proved before (with some few exceptions which we were aware of).

By starting out from the axioms of the real numbers we defined all the elementary functions including trigonometric functions and proved for these functions all the formulas which everyone knows from school. We have proved the Fundamental Theorem of Algebra by using the following concepts: continuity and compactness in metric spaces, trigonometric form of complex numbers, and Taylor's formula for polynomials. In short, we have derived the Fundamental Theorem of Algebra from the axioms of real numbers.

We have proved the theorem about integration by change of variables for real-valued functions of n-variables. The proof is very long and goes through much of linear algebra (determinants of matrices and linear operators), involves the topological concept of paracompactness, makes use of partition of unity, and fundamentally relies on the construction of the Lebesgue measure and a generalized version of Fubini's theorem. Naturally, the proof requires advanced tools of analysis to cope with Jocobians and the differentiability of functions of n-variables. We have all the components put together in our notebooks.

We have derived Zorn's Lemma from the Axiom of Choice and vice versa. We are aware where in mathematics the Axiom of Choice is used because we have paid attention to this matter as we wrote our proofs. On the pages of ProvenMath we are going to indicate whether a theorem uses the Axiom of Choice in its proof. We have already published on ProvenMath the theorem which establishes the equivalence of Zorn's Lemma and the Axiom of Choice.

In our research we have found it useful to apply the techniques of nets and subnets in topological spaces. In particular, we have proved and used the theorem that a topological space is compact if and only if every net contained in this space has a convergent subnet.

In our notebooks we have gathered solid knowledge of the fundaments of Real Analysis, Linear Algebra, Functional Analysis, Measure Theory, Probability Theory and General Topology. The existence of our hand-written proofs in our notebooks gives us certainty that it is possible to present this knowledge on the Internet in the form of theorems with complete proofs where every piece of knowledge can be traced down to the foundations of mathematics. This is what the ProvenMath Project is all about. We are going to gradually transfer our hand-written notes into the content of ProvenMath on the Internet.

If you want to help the ProvenMath Project you can contact us through the Apronus contact page. We are the authors and owners of the whole Apronus.com website. ProvenMath is just one of our ideas for the big Apronus project - honest stuff on the Internet.