Proof: Here we provide a proof using the maximum root. The left side of equation (13) can be seen as the maximum root of the following 8th-degree equation in T T T F ( T ) = ∏ c y c ( T ± x 2 + y z ± y 2 + z x ± z 2 + x y ) = T 8 − 4 ( ∑ c c c x 2 + ∑ c y c y z ) T 6 + 2 ( 3 ∑ c y c x 4 + 2 ∑ c y c x 3 ( y + z ) + 5 ∑ c c c y 2 z 2 + 8 ∑ c y c x 2 y z ) T 4 − 4 ( ∑ c y c x 6 − ∑ c c c x 5 ( y + z ) − 2 ∑ c c c x 4 ( y 2 + z 2 ) + 13 ∑ c c c x 4 y z + 11 ∑ c y c y 3 z 3 − 5 ∑ c c c x 3 ( y 2 z + y z 2 ) + 29 x 2 y 2 z 2 ) T 2 + ( ∑ c y c x 4 − 2 ∑ c y c x 3 ( y + z ) − ∑ c y c y 2 z 2 ) 2 = t 4 − 4 ( ∑ c c c x 2 + ∑ c y c y z ) t 3 + 2 ( 3 ∑ c y c x 4 + 2 ∑ c c x 3 ( y + z ) + 5 ∑ c y c y 2 z 2 + 8 ∑ c y c x 2 y z ) t 2 − 4 ( ∑ c y c x 6 − ∑ c y c x 5 ( y + z ) − 2 ∑ c c c x 4 ( y 2 + z 2 ) + 13 ∑ c y c x 4 y z + 11 ∑ c y c y 3 z 3 − 5 ∑ c y c x 3 ( y 2 z + y z 2 ) + 29 x 2 y 2 z 2 ) t + ( ∑ c y c x 4 − 2 ∑ c y c x 3 ( y + z ) − ∑ c y c y 2 z 2 ) 2 ≡ f ( t ) , \begin{aligned}
F(T)= & \prod_{\mathrm{cyc}}\left(T \pm \sqrt{x^{2}+y z} \pm \sqrt{y^{2}+z x} \pm \sqrt{z^{2}+x y}\right) \\
= & T^{8}-4\left(\sum_{\mathrm{ccc}} x^{2}+\sum_{\mathrm{cyc}} y z\right) T^{6}+2\left(3 \sum_{\mathrm{cyc}} x^{4}+2 \sum_{\mathrm{cyc}} x^{3}(y+z)\right. \\
& \left.+5 \sum_{\mathrm{ccc}} y^{2} z^{2}+8 \sum_{\mathrm{cyc}} x^{2} y z\right) T^{4}-4\left(\sum_{\mathrm{cyc}} x^{6}-\sum_{\mathrm{ccc}} x^{5}(y+z)\right. \\
& -2 \sum_{\mathrm{ccc}} x^{4}\left(y^{2}+z^{2}\right)+13 \sum_{\mathrm{ccc}} x^{4} y z+11 \sum_{\mathrm{cyc}} y^{3} z^{3} \\
& \left.-5 \sum_{\mathrm{ccc}} x^{3}\left(y^{2} z+y z^{2}\right)+29 x^{2} y^{2} z^{2}\right) T^{2} \\
& +\left(\sum_{\mathrm{cyc}} x^{4}-2 \sum_{\mathrm{cyc}} x^{3}(y+z)-\sum_{\mathrm{cyc}} y^{2} z^{2}\right)^{2} \\
= & t^{4}-4\left(\sum_{\mathrm{ccc}} x^{2}+\sum_{\mathrm{cyc}} y z\right) t^{3}+2\left(3 \sum_{\mathrm{cyc}} x^{4}+2 \sum_{\mathrm{cc}} x^{3}(y+z)\right. \\
& \left.+5 \sum_{\mathrm{cyc}} y^{2} z^{2}+8 \sum_{\mathrm{cyc}} x^{2} y z\right) t^{2}-4\left(\sum_{\mathrm{cyc}} x^{6}-\sum_{\mathrm{cyc}} x^{5}(y+z)\right. \\
& -2 \sum_{\mathrm{ccc}} x^{4}\left(y^{2}+z^{2}\right)+13 \sum_{\mathrm{cyc}} x^{4} y z+11 \sum_{\mathrm{cyc}} y^{3} z^{3} \\
& \left.-5 \sum_{\mathrm{cyc}} x^{3}\left(y^{2} z+y z^{2}\right)+29 x^{2} y^{2} z^{2}\right) t \\
& +\left(\sum_{\mathrm{cyc}} x^{4}-2 \sum_{\mathrm{cyc}} x^{3}(y+z)-\sum_{\mathrm{cyc}} y^{2} z^{2}\right)^{2} \\
\equiv & f(t),
\end{aligned} F ( T ) = = = ≡ cyc ∏ ( T ± x 2 + y z ± y 2 + z x ± z 2 + x y ) T 8 − 4 ( ccc ∑ x 2 + cyc ∑ y z ) T 6 + 2 ( 3 cyc ∑ x 4 + 2 cyc ∑ x 3 ( y + z ) + 5 ccc ∑ y 2 z 2 + 8 cyc ∑ x 2 y z ) T 4 − 4 ( cyc ∑ x 6 − ccc ∑ x 5 ( y + z ) − 2 ccc ∑ x 4 ( y 2 + z 2 ) + 13 ccc ∑ x 4 y z + 11 cyc ∑ y 3 z 3 − 5 ccc ∑ x 3 ( y 2 z + y z 2 ) + 29 x 2 y 2 z 2 ) T 2 + ( cyc ∑ x 4 − 2 cyc ∑ x 3 ( y + z ) − cyc ∑ y 2 z 2 ) 2 t 4 − 4 ( ccc ∑ x 2 + cyc ∑ y z ) t 3 + 2 ( 3 cyc ∑ x 4 + 2 cc ∑ x 3 ( y + z ) + 5 cyc ∑ y 2 z 2 + 8 cyc ∑ x 2 y z ) t 2 − 4 ( cyc ∑ x 6 − cyc ∑ x 5 ( y + z ) − 2 ccc ∑ x 4 ( y 2 + z 2 ) + 13 cyc ∑ x 4 y z + 11 cyc ∑ y 3 z 3 − 5 cyc ∑ x 3 ( y 2 z + y z 2 ) + 29 x 2 y 2 z 2 ) t + ( cyc ∑ x 4 − 2 cyc ∑ x 3 ( y + z ) − cyc ∑ y 2 z 2 ) 2 f ( t ) ,
where t = T 2 t=T^{2} t = T 2 t ⩾ 2 ( x + y + z ) 2 t \geqslant 2(x+y+z)^{2} t ⩾ 2 ( x + y + z ) 2 f ( t ) f(t) f ( t ) f ′ ( t ) 4 = t 3 − 3 ( ∑ c y c x 2 + ∑ c y c y z ) t 2 + ( 3 ∑ c y c x 4 + 2 ∑ c y c x 3 ( y + z ) + 5 ∑ c y c y 2 z 3 + 8 ∑ c y c x 2 y z ) t − ∑ c y c x 6 + ∑ c y c x 5 ( y + z ) + 2 ∑ c c c x 4 ( y 2 + z 2 ) − 13 ∑ c c t x 4 y z − 11 ∑ c v c y 3 z 3 + 5 ∑ c y c x 3 ( y 2 z + y z 2 ) − 29 x 2 y 2 z 2 = ( ι − 2 ( x + y + z ) 2 ) 3 + 3 ( ∑ c y c x 2 + 3 ∑ c y c y z ) ( t − 2 ( x + y + z ) 2 ) 2 ( 3 ∑ c y c x 4 + 14 ∑ c y c x 3 ( y + z ) + 29 ∑ c y c y 2 z 2 + 68 ∑ c y c x 2 y z ) ( t − 2 ( x + y + z ) 2 ) + ∑ c y c x 6 + 5 ∑ c y c x 5 ( y + z ) + 14 ∑ c y c x 4 ( y 2 + z 2 ) + 19 ∑ c y c x 4 y z + 9 ∑ c y c y 3 z 3 + 109 ∑ c y c x 3 ( y 2 z + y z 2 ) + 169 x 2 y 2 z 2 > 0 \begin{aligned}
\frac{f^{\prime}(t)}{4}= & t^{3}-3\left(\sum_{\mathrm{cyc}} x^{2}+\sum_{\mathrm{cyc}} y z\right) t^{2}+\left(3 \sum_{\mathrm{cyc}} x^{4}+2 \sum_{\mathrm{cyc}} x^{3}(y+z)\right. \\
& \left.+5 \sum_{\mathrm{cyc}} y^{2} z^{3}+8 \sum_{\mathrm{cyc}} x^{2} y z\right) t-\sum_{\mathrm{cyc}} x^{6}+\sum_{\mathrm{cyc}} x^{5}(y+z) \\
& +2 \sum_{\mathrm{ccc}} x^{4}\left(y^{2}+z^{2}\right)-13 \sum_{\mathrm{cct}} x^{4} y z-11 \sum_{\mathrm{cvc}} y^{3} z^{3} \\
& +5 \sum_{\mathrm{cyc}} x^{3}\left(y^{2} z+y z^{2}\right)-29 x^{2} y^{2} z^{2} \\
= & \left(\iota-2(x+y+z)^{2}\right)^{3}+3\left(\sum_{\mathrm{cyc}} x^{2}+3 \sum_{\mathrm{cyc}} y z\right)(t-2(x+y \\
& \left.+z)^{2}\right)^{2}\left(3 \sum_{\mathrm{cyc}} x^{4}+14 \sum_{\mathrm{cyc}} x^{3}(y+z)+29 \sum_{\mathrm{cyc}} y^{2} z^{2}\right. \\
& \left.+68 \sum_{\mathrm{cyc}} x^{2} y z\right)\left(t-2(x+y+z)^{2}\right)+\sum_{\mathrm{cyc}} x^{6}+5 \sum_{\mathrm{cyc}} x^{5}(y \\
& +z)+14 \sum_{\mathrm{cyc}} x^{4}\left(y^{2}+z^{2}\right)+19 \sum_{\mathrm{cyc}} x^{4} y z+9 \sum_{\mathrm{cyc}} y^{3} z^{3} \\
& +109 \sum_{\mathrm{cyc}} x^{3}\left(y^{2} z+y z^{2}\right)+169 x^{2} y^{2} z^{2}>0
\end{aligned} 4 f ′ ( t ) = = t 3 − 3 ( cyc ∑ x 2 + cyc ∑ y z ) t 2 + ( 3 cyc ∑ x 4 + 2 cyc ∑ x 3 ( y + z ) + 5 cyc ∑ y 2 z 3 + 8 cyc ∑ x 2 y z ) t − cyc ∑ x 6 + cyc ∑ x 5 ( y + z ) + 2 ccc ∑ x 4 ( y 2 + z 2 ) − 13 cct ∑ x 4 y z − 11 cvc ∑ y 3 z 3 + 5 cyc ∑ x 3 ( y 2 z + y z 2 ) − 29 x 2 y 2 z 2 ( ι − 2 ( x + y + z ) 2 ) 3 + 3 ( cyc ∑ x 2 + 3 cyc ∑ y z ) ( t − 2 ( x + y + z ) 2 ) 2 ( 3 cyc ∑ x 4 + 14 cyc ∑ x 3 ( y + z ) + 29 cyc ∑ y 2 z 2 + 68 cyc ∑ x 2 y z ) ( t − 2 ( x + y + z ) 2 ) + cyc ∑ x 6 + 5 cyc ∑ x 5 ( y + z ) + 14 cyc ∑ x 4 ( y 2 + z 2 ) + 19 cyc ∑ x 4 y z + 9 cyc ∑ y 3 z 3 + 109 cyc ∑ x 3 ( y 2 z + y z 2 ) + 169 x 2 y 2 z 2 > 0
Therefore, when t ⩾ 2 ( x + y + z ) 2 t \geqslant 2(x+y+z)^{2} t ⩾ 2 ( x + y + z ) 2 f ( t ) f(t) f ( t ) 9 4 ( x + y + z ) 2 > 2 ( x + y + z ) 2 , 256 f ( 9 4 ( x + y + z ) 2 ) = 625 ∑ cyc x 8 + 3800 ∑ cyc x 7 ( y + z ) + 14876 ∑ cyc x 6 ( y 2 + z 2 ) − 1912 ∑ cyc x 6 y z − 5400 ∑ cyc x 5 ( y 3 + z 3 ) + 48472 ∑ c y c x 5 ( y 2 z + y z 2 ) − 27802 ∑ cyc y 4 z 4 + 146248 ∑ cyc x 4 ( y 3 z + y z 3 ) + 289188 ∑ cyc x 4 y 2 z 2 + 490960 ∑ cyc x 2 y 3 z 3 = ( 25 ∑ cyc x 4 + 76 ∑ cyc x 3 ( y + z ) − 202 ∑ cyc y 2 z 2 − 270 ∑ cyc x 2 y z ) 2 + 19200 ∑ cyc x 6 ( y 2 + z 2 ) + 21504 ∑ cyc x 5 ( y 3 + z 3 ) − 81408 ∑ cyc y 4 z 4 + x y z ( 36 ∑ cyc x 5 + 133716 ∑ cyc x 4 ( y + z ) + 62856 ∑ cyc x 3 ( y 2 + z 2 ) ) + x 2 y 2 z 2 ( 226860 ∑ cyc x 2 + 285936 ∑ cyc y z ) \begin{array}{l}
\frac{9}{4}(x+y+z)^{2}>2(x+y+z)^{2}, \\
256 f\left(\frac{9}{4}(x+y+z)^{2}\right) \\
=625 \sum_{\text {cyc }} x^{8}+3800 \sum_{\text {cyc }} x^{7}(y+z)+14876 \sum_{\text {cyc }} x^{6}\left(y^{2}+z^{2}\right) \\
-1912 \sum_{\text {cyc }} x^{6} y z-5400 \sum_{\text {cyc }} x^{5}\left(y^{3}+z^{3}\right)+48472 \sum_{\mathrm{cyc}} x^{5}\left(y^{2} z\right. \\
\left.+y z^{2}\right)-27802 \sum_{\text {cyc }} y^{4} z^{4}+146248 \sum_{\text {cyc }} x^{4}\left(y^{3} z+y z^{3}\right) \\
+289188 \sum_{\text {cyc }} x^{4} y^{2} z^{2}+490960 \sum_{\text {cyc }} x^{2} y^{3} z^{3} \\
=\left(25 \sum_{\text {cyc }} x^{4}+76 \sum_{\text {cyc }} x^{3}(y+z)-202 \sum_{\text {cyc }} y^{2} z^{2}-270 \sum_{\text {cyc }} x^{2} y z\right)^{2} \\
+19200 \sum_{\text {cyc }} x^{6}\left(y^{2}+z^{2}\right)+21504 \sum_{\text {cyc }} x^{5}\left(y^{3}+z^{3}\right) \\
-81408 \sum_{\text {cyc }} y^{4} z^{4}+x y z\left(36 \sum_{\text {cyc }} x^{5}+133716 \sum_{\text {cyc }} x^{4}(y+z)\right. \\
\left.+62856 \sum_{\text {cyc }} x^{3}\left(y^{2}+z^{2}\right)\right)+x^{2} y^{2} z^{2}\left(226860 \sum_{\text {cyc }} x^{2}\right. \\
\left.+285936 \sum_{\text {cyc }} y z\right)
\end{array} 4 9 ( x + y + z ) 2 > 2 ( x + y + z ) 2 , 256 f ( 4 9 ( x + y + z ) 2 ) = 625 ∑ cyc x 8 + 3800 ∑ cyc x 7 ( y + z ) + 14876 ∑ cyc x 6 ( y 2 + z 2 ) − 1912 ∑ cyc x 6 y z − 5400 ∑ cyc x 5 ( y 3 + z 3 ) + 48472 ∑ cyc x 5 ( y 2 z + y z 2 ) − 27802 ∑ cyc y 4 z 4 + 146248 ∑ cyc x 4 ( y 3 z + y z 3 ) + 289188 ∑ cyc x 4 y 2 z 2 + 490960 ∑ cyc x 2 y 3 z 3 = ( 25 ∑ cyc x 4 + 76 ∑ cyc x 3 ( y + z ) − 202 ∑ cyc y 2 z 2 − 270 ∑ cyc x 2 y z ) 2 + 19200 ∑ cyc x 6 ( y 2 + z 2 ) + 21504 ∑ cyc x 5 ( y 3 + z 3 ) − 81408 ∑ cyc y 4 z 4 + x y z ( 36 ∑ cyc x 5 + 133716 ∑ cyc x 4 ( y + z ) + 62856 ∑ cyc x 3 ( y 2 + z 2 ) ) + x 2 y 2 z 2 ( 226860 ∑ cyc x 2 + 285936 ∑ cyc y z ) ⩾ 0 \geqslant 0 ⩾ 0
Therefore, equation (13) holds.