Some Properties of Real Maps

One can prove the following four propositions: (1) For all points p, q of E2 T and for every subset P of E2 T such that P is an arc from p to q holds P is compact. (2) For every real number r holds 0 ¬ r and r ¬ 1 iff r ∈ the carrier of I. (3) For all points p1, p2 of E 2 T and for all real numbers r1, r2 such that (1 − r1) · p1 + r1 · p2 = (1 − r2) · p1 + r2 · p2 holds r1 = r2 or p1 = p2. (4) Let p1, p2 be points of E 2 T . Suppose p1 6= p2. Then there exists a map f from I into (E2 T )↾L(p1, p2) such that for every real number x such that x ∈ [0, 1] holds f(x) = (1− x) · p1 + x · p2 and f is a homeomorphism and f(0) = p1 and f(1) = p2.