Please prove the following theorems using the provided axioms and definitions, using terms like suppose, let..ect. Please WRITE CLEARLY AND TYPE IF YOU CAN. 1 Order Properties Undefined Terms: The word "point" and the expression "the point x precedes the point y" will not be defined. This undefined expression will be written x 〈 y. Its negation, "x does not precede y," will be written X 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 x follows the point y', we mean y 〈 x. By x-y we mean that either x 〈 y or x = y. Also, x - 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 x < y or y <:x Axiom 2 (Axiom of point x precedes the point y, then x is not equal to y Axiom 3 (Axiom of and z are points such that x 〈 y and y , then x < z. ). S is a collection of points such that if x ). S is a collection of points such that if the ). S is a collection of points such that if x, y, 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 x and y. A region is a point set R such that there are points a and 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) Definition 4. A point set H is said to be open if for every point x of H there is a region R containing x such that RCH Definition 5. Two point sets are said to be mutually exclusive or disjoint if they have no points in common. If G is a collection of point sets such that each distinct pair of them are disjoint sets, then the sets of G are said to be pairwise disjoint. Definition 6. A point p is said to be a limit point of a point set M if and only if every region containing p contains a point of M distinct from p. The set of all limit points of a point set M is denoted M" Theorem 17. If p is a limit point of the union of a non-empty, finite collection G of point sets, then p is a limit point of at least one member of G 1 Order Properties Undefined Terms: The word "point" and the expression "the point x precedes the point y" will not be defined. This undefined expression will be written x 〈 y. Its negation, "x does not precede y," will be written X 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 x follows the point y', we mean y 〈 x. By x-y we mean that either x 〈 y or x = y. Also, x - 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 x