Two cyclic cubic sums bound a product of pairwise sums

x, y, z > 0, prove that;

$\left( x{y}^{2}+y{z}^{2}+z{x}^{2} \right) \left( {x}^{2}y+{y}^{2}z+{z}^{2}x \right) \geq \left( xy+zx+yz \right) \left( {x}^{2}{y}^{2}+{y}^{2}{z}^{2}+{x}^{2}{z}^{2} \right)$

Source: InternLM Lean-Workbook, record lean_workbook_10503 (Apache-2.0). Complete source proposition preserved; proof developed independently.