validation of a pumping lemma proof for regular languages


i have the following regular expression:

image of the regular expression

of course i could think of a world like w=a^(m+2)b^(m+2)c^(2m+3) and continue with the proof BUT i was just wondering, because L is made up of a union of two expressions, is it valid to split L into L1=a^(i)b^(j)c^(2j-1)| i<=j & i,j>0 L2=a^(i)b^(j)c^(2j-1)| i>=2j-1 & i,j>0

show for each L1 and L2 that the pummping lemma does not work on them (so for l1 i just show that for lets say each k i is not smaller or equal to j, and for l2 i show that i is not always bigger or equal to 2j-1 for each k)

by that i show that l1 and l2 are not regular which means that the union of the two will also be not regular.. is this corrent?

thank you.