Axioms of real numbers

3. Axiom of addition and multiplication:

4. Axioms of order:

4.1. x ≤ x

4.2. If x ≤ y and y ≤ x, then x = y

4.3. If x ≤ y and y ≤ z, then x ≤ z

4.4. If x ≤ y, then x + z ≤ y + z,

4.5. If θ ≤ x, θ ≤ y, then θ ≤ x ⋅ y

5. Axiom of completeness: Let X, Y and Z be subsets of R such, that and , it is known that and
it is valid y ≤ z. Then ∃ α ∈ X, for which