Properties of the residuum of a t-norm Let $\Rightarrow:[0,1]^2\to[0,1]$ such that $$ x\Rightarrow y=1\text{ iff }x\leq y $$ and $$ x\Rightarrow y=y\text{ iff }x>y $$ How can I prove, that $x\Rightarrow(y\Rightarrow z)=y\Rightarrow(x\Rightarrow z)$? Note: $\Rightarrow$ is the residuum of the Goedel t-norm.

The only way I can think of is to break into several cases based on the relative order of $x,y,z$. Not too many, though.

First assume $y\le z$. Then the LHS is $x\Rightarrow 1=1$. For the RHS, $x\Rightarrow z$ is equal to either $1$ or $z$. No matter which, we have $y \le (x\Rightarrow z)$, so the RHS is also $1$.

Next, assume $y>z$. Then the LHS becomes $x\Rightarrow z$, and you must break things up based on how $x$ compares to $z$. I'll leave the rest to the reader.