\(\dfrac{45}{19}-\left(\dfrac{1}{2}+\left(\dfrac{1}{3}+\left(\dfrac{1}{4}\right)^{-1}\right)^{-1}\right)^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{1}{2}+\left(\dfrac{1}{3}+4\right)^{-1}\right]^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{1}{2}+\left(\dfrac{13}{3}\right)^{-1}\right]^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{1}{2}+\dfrac{3}{13}\right]^{-1}\)
\(=\dfrac{45}{19}-\left(\dfrac{19}{26}\right)^{-1}\)
\(=\dfrac{45}{19}-\dfrac{26}{19}=\dfrac{19}{19}=1\)
\(\dfrac{45}{19}-\left[\dfrac{1}{2}+\left(\dfrac{1}{3}+\left(\dfrac{1}{4}\right)^{-1}\right)^{-1}\right]^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{1}{2}+\left(\dfrac{1}{3}+4\right)^{-1}\right]^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{1}{2}+\left(\dfrac{13}{3}\right)^{-1}\right]^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{1}{2}+\dfrac{3}{13}\right]^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{13}{26}+\dfrac{6}{26}\right]^{-1}\)
\(=\dfrac{45}{19}-\left[\dfrac{19}{26}\right]^{-1}\)
\(=\dfrac{45}{19}-\dfrac{26}{19}\)
\(=1\)
