1) On the basis of axioms 
• (adv.) By the use of axioms; in the form of an axiom.
Ax`i·o·mat'ic·al·ly adverb By the use of axioms; in the form of an axiom. 
[adv] - on the basis of axioms
axiomatical, axiomatically 1. Having the nature of an axiom; self-evident; characterized by axioms. 2. By the use of axioms; in the form of an axiom; such as, a statement or principle which is generally accepted to be true, but is not necessarily so. 3. A formal statement or principle in mathematics, science, etc., from which other statements can...
adverb on the basis of axioms; `this is axiomatically given`