# Invariance of $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$

I have to test whether the following system is invariant or not: $$T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$$, so I want to verify that, if $$y[n]=T[x[n]]$$, then $$y[n-N]=T[x[n-N]]$$. $$T[x[n-N]]=\sum_{k=n-1}^{n+2}x[k-N]$$ $$y[n-N]=\sum_{k=n-N-1}^{n-N+2}x[k]$$ In the first equation, if $$k'=k-N$$, we have that $$T[x[n-N]]=\sum_{k'=n-N-1}^{n-N+2}x[k']$$, so both sums are equal and thus the system is invariant. Is this correct or am I making wrong assumptions? Thanks in advance!

$$y[n] = T\{x[n]\} = x[n-1] + x[n] + x[n+1] + x[n+2] .$$
$$h[n] = \delta[n-1] + \delta[n] + \delta[n+1] + \delta[n+2]$$