Question

can you please prove the following theorem using the provided axioms and defintions. using terms like suppose in a paragraph format. please write clearly or type if you can !
1 Order Properties Undefined Terms: The word "point and the expression "the point z precedes the point y will not be defined. This undefined expression wil be written z < y. Its negation, "z does not precede y," will be written y. There is a set of all points, called the universal set, which is denoted S In these notes, all point sets have at least one element. By "point r follows the point y" we mean y < z. Byz ywe mean that either z < y or x-y Also,y means the points x and y are the same points, hence not different points. Axiom 1 (Axiom of and y are different points, then <y or y< r Axiom 2 (Axiom of point r precedes the point y, then is not equal to y Axiom 3 (Axiom of and z are points such that < y and y <z,then <z. . S is a collection of points such that if r S is a collection of points such that if the . S is a collection of points such that if x, y, In case a misunderstanding persists concerning the relationship between point," "point set," and S, let it be clear that, unless there is a stipulation to the contrary, that point" always means a member of S . "point set" means a nonempty set of points in S or a nonempty subset of .S is the universal set, that is, the set of all points. When we ascribe meanings for the undefined terms of our axiom system such that the axioms of our system are satisfied, we have a model of the axiom system thank you ! Axiom 4 (Axiom of. S has no first point and no last point Definition 1. If c is a point of a point set M such that no point of M precedes c, then c is called a first point of M. Similarly, if c is a point of a point set M such that then c is called a last point of M. Definition 2 (Spring 2019). A point d is an initial point of a set M if and only if all other points of M follow d. Definition 3. If z is a point such that x <z<y, then z is said to be between r and y. A region is a point set R such that there are points a and b where R- (x a <x<b}. We can represent the region of points between a and b by the notation (a, b). We will call a and b the of region (a, b).

Answer 1

Let, x elementof U_G elementof g G. Then, x elementof G_0 for some G_0 elementof g rightwardsdoublearrow Exists region R such that x elementof R Subset G_0 rightwardsdoublearrow x elementof R Subset U_G elementof g G rightwardsdoublearrow U_G elementof g G is open