Please prove the following indistinguishability properties. Please be detailed

Definition 4.5 Let Lleft and Lright be two libraries with a common interface. We say that Lleft and Lright (Indistinguishable) are indistinguishable, and write Left Lright, if for all polynomial-time programs A that output a single bit, Pr[A Lleft = 1] Pr[A. Lright = 1). We call the quantity | Pr[A Lleft = 1) Pr[A Lright = 1)| the advantage or bias of A in distinguishing Lleft from Lright. Two libraries are therefore indistinguishable if all polynomial-time calling programs have negligible advantage in distinguishing them. From the properties of the "z" symbol, we can see that indistinguishability of libraries is also transitive, which allows us to carry out hybrid proofs of security in the same way as before. If L1 = L2 then L, L2. Also, if L1 L2 L3 then L, L3. If Lleft Lright then L* Lleft L* . Lright for any polynomial-time library L. Definition 4.5 Let Lleft and Lright be two libraries with a common interface. We say that Lleft and Lright (Indistinguishable) are indistinguishable, and write Left Lright, if for all polynomial-time programs A that output a single bit, Pr[A Lleft = 1] Pr[A. Lright = 1). We call the quantity | Pr[A Lleft = 1) Pr[A Lright = 1)| the advantage or bias of A in distinguishing Lleft from Lright. Two libraries are therefore indistinguishable if all polynomial-time calling programs have negligible advantage in distinguishing them. From the properties of the "z" symbol, we can see that indistinguishability of libraries is also transitive, which allows us to carry out hybrid proofs of security in the same way as before. If L1 = L2 then L, L2. Also, if L1 L2 L3 then L, L3. If Lleft Lright then L* Lleft L* . Lright for any polynomial-time library L