The following is a part from P.13 in Topological Vector Spaces (Third Printing Corrected 1971) by H.H.Schaefer
Given a vector space $L$ over a (not necessarily commutative) non-discrete valuated field $K$ and a topology ${\mathfrak T}$ on $L$, the pair $(L,{\mathfrak T})$ is called a topological vector space (abbreviated t.v.s.) over $K$ if these two axioms are satisfied :
$(LT)_1 \;\; (x,y) \rightarrow x+y \;\; \mbox{is continuous on} \;\; L \times L \;\; \mbox{into} \;\; L $ $(LT)_2 \;\; (\lambda,x) \rightarrow \lambda x \;\; \mbox{is continuous on} \;\; K \times L \;\; \mbox{into} \;\; L $
Then, 3 lines below follows the following sentence.
Since, in particular, this implies the continuity of $(x,y) \rightarrow x-y$, every t.v.s. is a commutative topological group.
Isn't any vector space commutative by definition ? Or, is it about the commutativity of $K$ ? But then, how is it shown from the continuity of the mapping ? I'm confused.