Some Lemmas for the Jordan Curve Theorem 1 Andrzej Trybulec University of Białystok

The scheme NonEmpty deals with a non empty set A and a unary functor F yielding a set, and states that: {F(a) : a ranges over elements of A} is non empty for all values of the parameters. One can prove the following propositions: (1) For all sets A, B, C such that A ⊆ B and A misses C holds A ⊆ B \ C. (2) For all sets X, Y such that X meets ⋃ Y there exists a set Z such that Z ∈ Y and X meets Z. (3) For all sets A, B and for every function f such that A ⊆ dom f and fA ⊆ B holds A ⊆ f(B). (4) For every function f and for all sets A, B such that A misses B holds f(A) misses f(B).