answer please$-$ Theorem ex$7$:

$"$b$">>("$a$"$ U $"$b$"$ U Nil$)*>>$ Nil $=="$b$">>("$b$"$ U $"$a$")*.$

Proof.

Admitted.

Theorem ex$8$:

$(("$b$">>("$a$"$ U $\{\}))$ U $($Nil $>>\{\}>>"$c$")*)*==("$b$">>"$a$")*.$

Proof.

Admitted.