Integral of Measurable Function

One can prove the following propositions: (1) For all extended real numbers x, y holds |x− y| = |y − x|. (2) For all extended real numbers x, y holds y − x ≤ |x− y|. (3) Let x, y be extended real numbers and e be a real number. Suppose |x − y| < e and x 6= +∞ or y 6= +∞ but x 6= −∞ or y 6= −∞. Then x 6= +∞ and x 6= −∞ and y 6= +∞ and y 6= −∞. (4) For all extended real numbers x, y such that for every real number e such that 0 < e holds x < y + R(e) holds x ≤ y. (5) For all extended real numbers x, y, t such that t 6= −∞ and t 6= +∞ and x < y holds x+ t < y + t. (6) For all extended real numbers x, y, t such that t 6= −∞ and t 6= +∞ and x < y holds x− t < y − t.