You need to prove distributive law of sets such that:

`Ann (BUC) = (AnnB)U(AnnC)`

You need to prove that (by definition U)

=> `(x in A ^^ x in B) vv (x in A ^^ x in C) ` (use distibution from logic equivalence)

=>`(x in Ann B) vv...

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You need to prove distributive law of sets such that:

`Ann (BUC) = (AnnB)U(AnnC)`

You need to prove that (by definition U)

=> `(x in A ^^ x in B) vv (x in A ^^ x in C) ` (use distibution from logic equivalence)

=>`(x in Ann B) vv (x in A nn C) ` (use definition of `nn` )

=>`(x in Ann B)U (x in A nn C)` (use definition of U).

**Hence, using logic equivalences yields the distributive law of sets such that A`nn` (BUC) `= (AnnB)U(AnnC)` .**