Show that holomorphic functions are infinitely differentiable without complex analysis 
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Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions? 

Is there a way to show that holomorphic functions, thought as real functions, are   $C^\infty$ , without using any Complex Analysis?
That is,
Let $f:A\rightarrow R^2$ where $A \subseteq R^2$ is open, f is differentiable on A, and satisfies the Cauchy Riemann Equations. Show that f is $C^\infty$ on $A$, without using any Complex Analysis definiton, theorem, or tool.
 A: It can be instructive to consider the Cauchy-Riemann equations as an elliptic system of partial differential equations. For elliptic systems, there is a $C^\infty$-regularity result similar to the one known for a single equation (see L. H\"ormander, Linear partial differential operators, Springer, 1963).
A: Edit: Avoided by $C^2$ by using integrals.
Edit 2: I believe that the following manipulations of $U,V$ work out right...
If $f$ satisfies the Cauchy-Riemann equations, then write $f(x,y) = (u(x,y), v(x,y))$. Now, let $U,V$ be the antiderivatives of $u,v$ with respect to $x$. Since $u,v \in C^1$, we claim that $U,V \in C^2$ and are harmonic. Using uniform convergence to exchange integral and derivative,
\begin{equation}
U_{xx} = D_x^2 \int u \ dx = D_x \int u_x \ dx = u_x
\end{equation}
\begin{equation}
U_{yy} = D_y^2 \int u \ dx = D_y \int u_y \ dx = D_y \int -v_x \ dx = -v_y = -u_x
\end{equation}
The computations for $V$ should be similar. Hence $U,V$ are harmonic.
A harmonic function is defined by satisfying the Laplace equation. We can prove that every real harmonic function is smooth by showing that the convolution of any harmonic function by a mollifier is in fact equal to the same harmonic function; this proves that the harmonic function is smooth. If you want a reference, see, for example, L. Craig Evans' textbook "Partial Differential Equations". 
A: This already got asked, and I answered it: Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?
