Assertions:
\(t \leq v+w\)
\(w+u \leq x+z\)
\(v+x \leq y\)
Conclusion: \(t+u \leq y+z\)
Proof:
\((t)+(u) \leq (v+w)+(u)\) - from monotonicity and reflexivity of u
\(= v+(w+u)\) - associativity
\(\leq v+(x+z)\) - monotonicity and reflexivity of v
\(= (v+x)+z\) - associativity
\(\leq y+z\) - monotonicity and reflexivity of z
Symmetry was not needed because the diagram had no crossing wires.