On Ordering of Bags

The following propositions are true: (1) For all sets x, y, z such that z ∈ x and z ∈ y holds x \ {z} = y \ {z} iff x = y. (2) For all natural numbers n, k holds k ∈ Seg n iff k−1 is a natural number and k − 1 < n. Let f be a natural-yielding function and let X be a set. One can verify that f↾X is natural-yielding. Let f be a finite-support function and let X be a set. One can check that f↾X is finite-support. Next we state three propositions: (3) For every function f and for every set x such that x ∈ dom f holds f · 〈x〉 = 〈f(x)〉.