@MISC{Białas_someproperties, author = {Józef Białas}, title = {Some Properties of the Intervals MML Identifier:MEASURE6.}, year = {} }

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Abstract

for this paper. The following propositions are true: (1) There exists a function F from N into [:N, N:] such that F is one-to-one and domF = N and rngF = [:N, N:]. (2) For every function F from N into R such that F is non-negative holds 0 R ≤ ∑F. (3) Let F be a function from N into R and x be an extended real number. Suppose there exists a natural number n such that x ≤ F(n) and F is non-negative. Then x ≤ ∑F. (8) 1 For all extended real numbers x, y such that x is a real number holds (y − x)+x = y and (y+x) − x = y. (10) 2 For all extended real numbers x, y, z such that z ∈ R and y < x holds (z+x)−(z+y) = x−y. (11) For all extended real numbers x, y, z such that z ∈ R and x ≤ y holds z + x ≤ z + y and x+z ≤ y+z and x − z ≤ y − z. (12) For all extended real numbers x, y, z such that z ∈ R and x < y holds z + x < z + y and x+z < y+z and x − z < y − z. Let x be a real number. The functor R(x) yielding an extended real number is defined as follows: (Def. 1) R(x) = x. We now state a number of propositions: (13) For all real numbers x, y holds x ≤ y iff R(x) ≤ R(y). (14) For all real numbers x, y holds x < y iff R(x) < R(y). (15) For all extended real numbers x, y, z such that x < y and y < z holds y is a real number. (16) Let x, y, z be extended real numbers. Suppose x is a real number and z is a real number and x ≤ y and y ≤ z. Then y is a real number. (17) For all extended real numbers x, y, z such that x is a real number and x ≤ y and y < z holds y is a real number.