Exercise $4$ Consider the following four formulas over the vocabulary LA:

P$1$: $$x$($sx $6=0)$

P$2$: $$x$$y$($sx $=$ sy $$ x $=$ y$)$

P$3$: $$x$($x $+0=$ x$)$

P$4$: $$x$$y$($x $+$ sy $=$ s$($x $+$ y$))$

Prove from the definition of $|=$ that

P$1,$P$2,$P$3,$P$46|=$x$$y$($x $+$ y $=$ y $+$ x$)$

Hint: Think of $+$ as string concatenation