$K$ = $\frac{1}{2}mv^2$

$\frac{dK}{dt}$= $mv \frac{dv}{dt}$=$mv a_t$=$(ma_t)v$=$F_t v$

$dK$ = $F_t vdt$ = $F_t (vdt)$ = $\overrightarrow{F_t} .\overrightarrow{dr}$

$\triangle K$ = $W$_all

$\triangle U$ = $-W$_{internal, conservative}

If $\overrightarrow{F}$_external=$0$ and $\overrightarrow{F}$_{internal, non-conservative}=$0$

$\triangle K$ = $W$_all = $W$_{internal, conservative} ($\because$ $W$_{internal, non-conservative} and $W$_external = 0)

$\Rightarrow$ $\triangle K$ = $- \triangle U$

$\Rightarrow$ $K_f - K_i$ = $U_i - U_f$

$\Rightarrow$ $K_f + U_f$ = $K_i + U_i$

Thus, principal of conservation of energy holds true in the given conditions.

$W$_all=$\triangle K$

$\Rightarrow$ $W$_{internal, conservative}+$W$_{internal, non-conservative} + $W$_external = $K_f$ - $K_i$

$\Rightarrow$ ($U_i$ - $U_f$) + $W$_{internal, non-conservative} + $W$_external = $K_f$ - $K_i$

$\Rightarrow$ $W$_{internal, non-conservative} + $W$_external = ($K_f$ + $U_f$) - ($K_i$ + $U_i$)

$\Rightarrow$ $W$_{internal, non-conservative} + $W$_external = $E$_{total final} - $E$_{total initial}

$\Rightarrow$ if $W$_{internal, non-conservative}= $0$, then $W$_external = $\triangle E$_total