Let me do the first ones.

Definition:Let then we define iff .

Theorem:Let then .

Proof:We need to show for some . So choose .Q>E>D>

Theorem:Let then .

Proof:We need to show for some . So choose .Q.E.D.

Theorem:Let then .

Proof:We need to show for some . So chose .Q.E.D.

Theorem:Let and then .

Proof:We need to find for some . If then which is impossible. If it is impossible. So the only possible case is .Q.E.D.