Could someone please check my work

Prove Theorem: Ler A be a subset of U. Then AU ( U( A) = U. Fill in The blanks. Preof: If X E AU [ UV A), Then XE A or XE UJ A. Since both A and UJA are subsets of U. in either case we have XEU. Thus, AU(U J A) S U. On The other hand, suppose That REV. Now either XEA or XA. If KEA, Then XE ULA. In either Case XE AU ( UV A). Hence U = AU ( UJA)