Is this method of solving equations true?

Question

Is this statement true , for all numbers " If A * B * C = X * Y * Z , THEN {X,Y,Z}={A,B,C} " ? If not , when is this statement true ? I ask this question as I have came across the solution of a system of equations in a textbook , it concludes with As (Z-x)(Z-y)(Z-z) = (Z-a)(Z-b)(Z-c) , then (x,y,z) is a permuation on {a,b,c} . ( Z (capital Z) is a a variable in the set of complex numbers, we solve for (x,y,z) )