I have dealt with the following question from the theory of dimension in commutative algebra.

To let $ (A, m) $ be a local ring and $ M $ a finally generated $ A $-Module.

given $ x_1, …, x_r in M $, Prove that $ dim ( frac {M} {(x_1, …, x_r) M}) geq dim (M) – r $with equality, if and only if {$ x_1, …, x_r $} is part of a parameter system for $ M $,

Now I can show that, though $ A $ is a $ mathbf {regular} $ local ring so $ frac {A} {(x_1, …, x_r) A} $ is a regular local ring with dimension $ dim (A) – r $ then and only if {$ x_1, …, x_r $} is part of a parameter system for $ A $, But I do not know how to show that for the given case. I also can not show the inequality. I could not find any proof, so I would be grateful for any help!