The following problem is an exercise in Loday-Vallette's Algebraic Operads. I hope I am understanding this correctly. Any suggestions or hints would be appreciated.
Show that the restriction of the product $\mu$ and the reduced coproduct $\Delta$ on the augmentation ideal $\bar{H}$ of a bialgebra $H$ satisfy the following relation: $$\Delta( \mu(x\otimes y))=x\otimes y+y\otimes x+(\text{id}\otimes \mu)(\Delta(x)\otimes y)+(\mu\otimes \text{id})(\text{id}\otimes \tau)(\Delta(x)\otimes y)+(\mu\otimes \text{id})(x\otimes \Delta(y))+(\text{id}\otimes \mu)(\tau\otimes \text{id})(x\otimes \Delta(y))+(\mu\otimes \mu)(\text{id}\otimes \tau\otimes \text{id})(\Delta(x)\otimes \Delta(y))$$ How does one even go about solving such a problem? Also what is confusing me, in 1.3 the definition of bialgebra, they say that the compatibility relation reads $$\Delta( \mu(x\otimes y))=(\mu\otimes \mu)(\text{id}\otimes \tau\otimes \text{id})(\Delta(x)\otimes \Delta(y))$$ so why are there other terms now? I don't know how to create all these new terms on the RHS. Thanks for the help.