@MISC{Warsaw_partialfunctions, author = {Czeslaw Bylinski Warsaw}, title = {Partial Functions}, year = {} }

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Abstract

this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following rules: x, y, y 1 , y 2 , z, z 1 , z 2 denote sets, P , Q, X, X # , X 1 , X 2 , Y , Y # , Y 1 , Y 2 , V , Z denote sets, and C, D denote non empty sets. We now state three propositions: (1) If P # [: X 1 , Y 1 :] and Q # [: X 2 , Y 2 :], then P # Q # [: X 1 # X 2 , Y 1 # Y 2 :]. (2) For all functions f , g such that for every x such that x # dom f # dom g holds f(x) = g(x) there exists a function h such that f # g = h. (3) For all functions f , g, h such that f # g = h and for every x such that x # dom f # dom g holds f(x) = g(x). The scheme LambdaC deals with a set A, a unary functor F yielding a set, a unary functor G yielding a set, and a unary predicate P, and states that: There exists a function f such that dom f = A and for every x such that x # A holds if P[x], then f(x) = F(x) and if not P[x], then f(x) = G(x) for all values of the parameters. One can check that there exists a function which is empty. The following proposition is true (10) 1 rng # = #. Let us consider X, Y . Observe that there exists a relation between X and Y which is function-like. Let us consider X, Y . A partial function from X to Y is a function-like relation between X and Y . One can prove the following propositions: 1 The propositions (4)--(9) have been removed. 1 c # Association of Mizar Users partial functions 2 (24) 2 Every function f is a partial function from dom f to rng f. (25) For every function f such that rng f # Y holds f i...