pumping lemma – Prove that L is not a Context-free language

Prove that L is not a Context-free language.
L = { $a^{i}$ $b^{j}$ $c^{h}$ | $i,j,h$ $in$ N and h = $max$($i,j$)}

I have an idea: It can be divided into two situations:

$1.$ when $i$ < $j$, $w$ = $a^{i}$ $b^{j}$ $c^{i}$

$2.$ when $i$ > $j$, $w$ = $a^{i}$ $b^{j}$ $c^{j}$

Then with the help of the pump lemma,but I will only use special examples to prove it. I always feel not rigorous. How should I write it more rigorously?

My writing is:

when $i$ < $j$, $w$ = $a^{i}$ $b^{j}$ $c^{i}$, i = 4, j = 3. $w$ = aaaa bbb cccc, then use uvxyz to prove step by step.