(2)1 For every real number a such that 1 ≤ a holds a ≤ a2. (3) For every real number a such that −1 ≥ a holds −a ≤ a2. (4) For every real number a such that −1 > a holds −a < a2. (5) For all real numbers a, b such that b2 ≤ a2 and a ≥ 0 holds −a ≤ b and b ≤ a. (6) For all real numbers a, b such that b2 < a2 and a ≥ 0 holds −a < b and b < a. (7) For all real numbers a, b such that −a ≤ b and b ≤...

In previous joint work with Eli Aljadeff we attached a generic Hopf Galois extension A H to each twisted algebra H obtained from a Hopf algebra H by twisting its product with the help of a cocycle α. The algebra A H is a flat deformation of H over a “big” central subalgebra B H and can be viewed as the noncommutative analogue of a versal torsor in the sense of Serre. After surveying the results...

One can prove the following propositions: (2) For all real numbers a, b, r such that 0 ¬ r and r ¬ 1 and a ¬ b holds a ¬ (1 − r) · a + r · b and (1 − r) · a + r · b ¬ b. (3) For all real numbers a, b such that a ­ 0 and b > 0 or a > 0 and b ­ 0 holds a + b > 0. (4) For all real numbers a, b such that −1 ¬ a and a ¬ 1 and −1 ¬ b and b ¬ 1 holds a · b ¬ 1. (5) For all real numbers a, b such that ...