  In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for
each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that
its triangular Walsh-Fourier series diverges almost everywhere. In this paper
we investigate the Fej\'er (or $(C,1)$) means of the triangle two variable
Walsh-Fourier series of $L^1$ functions. Namely, we prove the a.e. convergence
$\sigma_n^{\bigtriangleup}f = \frac{1}{n}\sum_{k=0}^{n-1}S_{k, n-k}f\to f$
($n\to\infty$) for each integrable two-variable function $f$.
