$a, b, c$ are ordinals. Ordinal addition is defined as $$a + b = type(\{0\}\times a\cup \{1\}\times b, L)$$ where L is the lexicographic ordering. Ordinal $a \leq b$ is defined as $a < b$ or $a = b$, where $<$ is defined by $\in$ relation.
I'm trying to prove that $a + c\leq b + c$ if $a < b$, but I'm having quite some trouble using the $\leq$ definition I was given here.