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Need some help with simple proofs watch

1. Hey, I'm currently constructing some simple proofs as an exercise, I was wondering if someone from TSR could give me a hand with them.

This is the first one, I will probably put more up.

Code:
Theory: A/(A/B)=B/(B/A)

Proof: Let x∊A/(A/B) then x∊A and x∉A/B, therefore x∊ and (x∉A or x∊B). Using the dissociative law we have two cases to consider either x∊ A and x∉A which leads to a contradiction so we need not consider it, or x∊A and  x∊B. If x∊A and  x∊B then x∉B/A. As  x∊B and x∉B/A then x∊B/(B/A) and thus A/(A/B)⊆B/(B/A).
Conversely let y∊B/(B/A) then y∊B and y∉B/A. Therefore we can once again reach the conclusion that y∊B and y∊A. Therefore y∉A/B and further y∊A/(A/B) and thus we conclude B/(B/A)⊆A/(A/B) as required.
2. For what it's worth, shouldn't those "/"s be "\"s?

I think your proof is alright, although probably slightly longer than it needs to be. You can also use the rules and to neaten it up.

[Also, avoid the [ code ] tag, it makes it harder to read!]

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Updated: September 29, 2010
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