Journal of Formalized Mathematics

The articles [19], [7], [22], [1], [20], [15], [13], [21], [9], [4], [6], [5], [3], [16], [12], [2], [17], [18], [10], [8], [11], and [14] provide the notation and terminology for this paper. We adopt the following rules: k, n are natural numbers, r, r′, r1 are real numbers, and c, c′, c1, c2 are elements of C. In this article we present several logical schemes. The scheme FuncDefUniq deals with non empty sets A , B and a unary functor F yielding a set, and states that: Let f1, f2 be functions from A into B. Suppose for every element x of A holds f1(x) = F (x) and for every element x of A holds f2(x) = F (x). Then f1 = f2 for all values of the parameters. The scheme BinOpDefuniq deals with a non empty set A and a binary functor F yielding a set, and states that: Let o1, o2 be binary operations on A . Suppose for all elements a, b of A holds o1(a, b) = F (a,b) and for all elements a, b of A holds o2(a, b) = F (a,b). Then o1 = o2 for all values of the parameters. The binary operation +C on C is defined as follows: