Representation of the Fibonacci and Lucas Numbers in Terms of Floor and Ceiling

One can prove the following propositions: (1) For all real numbers a, b and for every natural number c holds (ab ) c = a c bc . (2) For every real number a and for all integer numbers b, c such that a 6= 0 holds ab+c = ab · ac. (3) For every natural number n and for every real number a such that n is even and a 6= 0 holds (−a)n = an. (4) For every natural number n and for every real number a such that n is odd and a 6= 0 holds (−a)n = −an. (5) |τ | < 1. (6) For every natural number n and for every non empty real number r such that n is even holds rn > 0. (7) For every natural number n and for every real number r such that n is odd and r < 0 holds rn < 0.