Hahn-banach Theorem

We prove a version of Hahn-Banach Theorem. and 1] provide the notation and terminology for this paper. The following propositions are true: (1) For all sets x, y and for every function f such that h hx; yi i 2 f holds y 2 rng f: (2) For every set X and for all functions f, g such that X dom f and f g holds fX = gX: (3) For every non empty set A and for every set b such that A 6 = fbg there exists an element a of A such that a 6 = b: (4) For all sets X, Y holds every non empty subset of X _ !Y is a non empty functional set. (5) Let B be a non empty functional set and f be a function. Suppose f = S B: Then dom f = S fdomg : g ranges over elements of Bg and rng f = S frngg : g ranges over elements of Bg: (6) For every non empty subset A of R such that for every Real number r such that r 2 A holds r ?1 holds A = f?1g: (7) For every non empty subset A of R such that for every Real number r such that r 2 A holds +1 r holds A = f+1g: (8) Let A be a non empty subset of R and r be a Real number. If r < sup A; then there exists a Real number s such that s 2 A and r < s: (9) Let A be a non empty subset of R and r be a Real number. If inf A < r; then there exists a Real number s such that s 2 A and s < r: (10) Let A, B be non empty subsets of R. Suppose that for all Real numbers r, s such that r 2 A and s 2 B holds r s: Then sup A inf B: 1 c Association of Mizar Users