Cauchy's Theorem. Let R be a simply connected domain and f a function on R to E[squared] which is differentiable over R. Let C be any closed rectifiable curve in R. Then I [integral][lowerd C]f(z)dz = 0. Through the years Cauchy's Theorem was the only means of proving that if a complex valued function f(z) is differentiable over a domain R then f(z) has derivatives of all orders in R. A proof of the above statement without the use of an Integral was given for the first time in 1960 by H. E. Connell, The purpose of this thesis is to make a resume of all the theorems leading to Connell's proof.