Difference and Difference Quotient. Part IV

The papers [2], [7], [13], [3], [1], [6], [9], [4], [14], [8], [5], [15], [11], [12], and [10] provide the notation and terminology for this paper. We adopt the following rules: n denotes an element of N, h, k, x, x0, x1, x2, x3 denote real numbers, and f , g denote functions from R into R. Next we state a number of propositions: (1) If x0 > 0 and x1 > 0, then loge x0 − loge x1 = loge( x1 ). (2) If x0 > 0 and x1 > 0, then loge x0 + loge x1 = loge(x0 · x1). (3) If x > 0, then loge x = (the function ln)(x). (4) If x0 > 0 and x1 > 0, then (the function ln)(x0)−(the function ln)(x1) = (the function ln)(0 x1 ). (5) Suppose for every x holds f(x) = k x2 and x0 6= 0 and x1 6= 0 and x2 6= 0 and x3 6= 0 and x0, x1, x2, x3 are mutually different. Then ∆[f ](x0, x1, x2, x3) = k·( 1 x1·x2·x0 ·( 1 x0 + 1 x2 + 1 x1 )− 1 x2·x1·x3 ·( 1 x3 + 1 x1 + 1 x2 )) x0−x3 .