$\begin{array}{l} a + b + c = \sqrt a + \sqrt b + \sqrt c = 2\\ \Rightarrow 4 = {\left( {\sqrt a + \sqrt b + \sqrt c } \right)^2} = a + b + c + 2\sqrt {ab} + 2\sqrt {bc} + 2\sqrt {ca} \\ \Leftrightarrow 2 = 2\sqrt {ab} + 2\sqrt {bc} + 2\sqrt {ca} \Leftrightarrow \sqrt {ab} + \sqrt {bc} + \sqrt {ca} = 1 \end{array}$
$\begin{array}{l}
\sqrt {ab} + \sqrt {bc} + \sqrt {ca} = 1\\
1 + a = a + \sqrt {ab} + \sqrt {bc} + \sqrt {ca} = \sqrt a \left( {\sqrt a + \sqrt b } \right) + \sqrt c \left( {\sqrt a + \sqrt b } \right) = \left( {\sqrt a + \sqrt b } \right)\left( {\sqrt a + \sqrt c } \right)\\
1 + b = \left( {\sqrt b + \sqrt c } \right)\left( {\sqrt b + \sqrt a } \right)\\
1 + c = \left( {\sqrt c + \sqrt a } \right)\left( {\sqrt c + \sqrt b } \right)\\
\Rightarrow \sqrt {\left( {1 + a} \right)\left( {1 + b} \right)\left( {1 + c} \right)} = \left( {\sqrt a + \sqrt b } \right)\left( {\sqrt b + \sqrt c } \right)\left( {\sqrt c + \sqrt a } \right)\\
VT = \dfrac{{\sqrt a }}{{1 + a}} + \dfrac{{\sqrt b }}{{1 + b}} + \dfrac{{\sqrt c }}{{1 + c}}\\
VT = \dfrac{{\sqrt a }}{{\left( {\sqrt a + \sqrt b } \right)\left( {\sqrt a + \sqrt c } \right)}} + \dfrac{{\sqrt b }}{{\left( {\sqrt b + \sqrt c } \right)\left( {\sqrt b + \sqrt a } \right)}} + \dfrac{{\sqrt c }}{{\left( {\sqrt c + \sqrt a } \right)\left( {\sqrt c + \sqrt b } \right)}}\\
VT = \dfrac{{\sqrt a \left( {\sqrt b + \sqrt c } \right) + \sqrt b \left( {\sqrt a + \sqrt c } \right) + \sqrt c \left( {\sqrt a + \sqrt b } \right)}}{{\left( {\sqrt a + \sqrt b } \right)\left( {\sqrt b + \sqrt c } \right)\left( {\sqrt c + \sqrt a } \right)}}\\
VT = \dfrac{{\sqrt {ab} + \sqrt {ac} + \sqrt {ab} + \sqrt {bc} + \sqrt {ac} + \sqrt {ca} + \sqrt {cb} }}{{\left( {\sqrt a + \sqrt b } \right)\left( {\sqrt b + \sqrt c } \right)\left( {\sqrt c + \sqrt a } \right)}}\\
VT = \dfrac{{2\left( {\sqrt {ab} + \sqrt {bc} + \sqrt {ca} } \right)}}{{\left( {\sqrt a + \sqrt b } \right)\left( {\sqrt b + \sqrt c } \right)\left( {\sqrt c + \sqrt a } \right)}} = \dfrac{2}{{\left( {\sqrt a + \sqrt b } \right)\left( {\sqrt b + \sqrt c } \right)\left( {\sqrt c + \sqrt a } \right)}}\\
= \dfrac{2}{{\sqrt {\left( {1 + a} \right)\left( {1 + b} \right)\left( {1 + c} \right)} }} = VP\\
Q.E.D
\end{array}$
