Link to code executable online

rho[x1_, x2_, x3_, y1_, y2_, y3_] = x1^2*(y2 + y3) + x2^2*(y1 + y3) + x3^2*(y1 + y2) + 2*y1*y2*y3 + x1*x2*(y1 + y2 + 2*y3) + x1*x3*(y1 + 2*y2 + y3) + x2*x3*(2*y1 + y2 + y3) - x1*(y1*y2 + y1*y3 + 2*y2*y3) - x2*(y1*y2 + 2*y1*y3 + y2*y3) - x3*(2*y1*y2 + y1*y3 + y2*y3)

left = (x2 + x3 - 1)*(x1 + x3 - 1)*rho[x1, x2, x3, y1, y2, y3]

right = (x2 + x3 - x1 + y1)*(x1 + x3 - x2 + y2)*rho[x1, x2, x3 - 1, y1, y2, y3 - 1] + (x1 - y1)*(x1 + x3 - x2 + y2)*rho[x1, x2, x3 - 1, y1 + 1, y2, y3 - 1] + (x2 - y2)*(x2 + x3 - x1 + y1)*rho[x1, x2, x3 - 1, y1, y2 + 1, y3 - 1] + (x1 - y1)*(x2 - y2)*rho[x1, x2, x3 - 1, y1 + 1, y2 + 1, y3 - 1]

Print[
  FullSimplify[
    FunctionExpand[
	left==right
    ]
  ]
]