Let $K \equiv \lambda xy.x$ and $S \equiv \lambda xyz.xz(yz)$. Show that S and K are incompatible.
The solution goes like:
Let $S=K$ and $I \equiv \lambda x.x$, we have to show that all terms are equal. Since $S=K$ we have $SIMI=KIMI$ for any terms M.
$KIMI=(\lambda xy.x)(\lambda x.x)MI = (\lambda yx.x) (M)I=\lambda x.x I=II=I$ (I understand this.)
and
$SIMI=II(MI)=I(MI)=MI$ (This is the reduction I don't understand, specially the first step.)
Thus, we have I=MI. Now, let $M \equiv \lambda x.N$, then we have $I=N$ for all terms N, and we are done.
Thanks in advance!
The first step is just 3 one-step reductions, resulting in substituting $I$, $M$, and $I$ for $x$, $y$, and $z$ respectively into the definition of $S$: $$ \begin{array} & SIMI &= (\lambda xyz.xz(yz))\ I M I \quad & [\text{by definition of }S]\\ & = I I (M I) & [\text{3 arguments substituted}] \\ &= (I I) (M I) & [\text{implicitly understood parens added}] \\ &= I (M I) \\ &= M I \end{array} $$