The Fashoda Meet Theorem for Rectangles

The articles [1], [6], [15], [17], [5], [2], [3], [16], [7], [14], [13], [10], [11], [8], [4], [9], and [12] provide the notation and terminology for this paper. One can prove the following propositions: (1) For all real numbers a, b, d and for every point p of E2 T such that a < b and p2 = d and a ≤ p1 and p1 ≤ b holds p ∈ L([a, d], [b, d]). (2) Let n be a natural number, P be a subset of E T, and p1, p2 be points of E T. Suppose P is an arc from p1 to p2. Then there exists a map f from I into E T such that f is continuous and one-to-one and rng f = P and f(0) = p1 and f(1) = p2. (3) Let p1, p2 be points of E 2 T and b, c, d be real numbers. If (p1)1 < b and (p1)1 = (p2)1 and c ≤ (p1)2 and (p1)2 < (p2)2 and (p2)2 ≤ d, then p1 ≤Rectangle((p1)1,b,c,d) p2. (4) Let p1, p2 be points of E 2 T and b, c be real numbers. Suppose (p1)1 < b and c < (p2)2 and c ≤ (p1)2 and (p1)2 ≤ (p2)2 and (p1)1 ≤ (p2)1 and (p2)1 ≤ b. Then p1 ≤Rectangle((p1)1,b,c,(p2)2) p2. (5) Let p1, p2 be points of E 2 T and c, d be real numbers. Suppose (p1)1 < (p2)1 and c < d and c ≤ (p1)2 and (p1)2 ≤ d and c ≤ (p2)2 and (p2)2 ≤ d. Then p1 ≤Rectangle((p1)1,(p2)1,c,d) p2.