Summary Previous Mizar articles [7, 6, 5] formalized the implicit and inverse function theorems for Frechet continuously differentiable maps on Banach spaces. In this paper, using system [1], [2], we formalize these Euclidean spaces by specializing them. We referred to [4], [12], [10], [11] in formalization.

I A more general theory will result if in the place of R we employ an abstract normed ring. s We use the symbols =, ... ... in more than one sense. No confusion need arise as tie context makes clear the meaning of each such symbol. It is worth while to mention here that the relation of equality = for E1 as well as for E2 is not an independent primitive idea; for, an equivalent set of postulates...

We remark that the result presented here can be proved more simply using the closed graph theorem. However we believe that our results can be used to prove a more interesting result. Details to follow in a later paper. One of the remarkable features of the space of holomorphic functions (in either one or several variables) is that the standard Frechet space topologies— say, for example, the L o...