Fisher effect in tax competition

Hello,
am asking you Check to see if this is logical, true and correct computes


(1+ i)=(1+ ρ)(1+ π)

we have Fisher equation :

i=ρ+π+ρπ (1)

The FOC of firm's optimization problem in tax competition model of Zodrow and Mieszkowski (1986):

f'(k)= ρ+t (2)

On a :

ρ=i-π (3)

replace (3) in (2), we obtain :

f'(k)= i-π+t (4)

And replace (1) in (4), we obtain :

f'(k)=ρ+π+ρπ-π+t=ρ+ρπ+t=(1+π)ρ+t

⟹ f'(k)=(1+π)ρ+t (5)

Thank you