For similarly, given two Mealy machines, let (Me₁)(Me₂) mean that an input string is processed on Me₁ and then the output string is immediately fed into Me₂ (as input) and reprocessed. Only this second resultant output is considered the final output of (Me₁)(Me₂). If the final output string is the same as the original input string, we say that (Me₁)(Me₂) has the identity property, symbolically written (Me₁)(Me₂) = identity.
Given two specific machines such that (Me₁)(Me₂) reproduces the original bit string, we aim to prove (in the following two problems) that (Me₂)(Me₁) must necessarily also have this property.

Take the equality (Me₁)(Me₂) = identity. Multiply both sides by Me₁ to get (Me₁) (Me₂)(Me₁) = identity (Me₁) = Me₁ . This means that (Me₂)(Me₁) takes all outputs from Me₁ and leaves them unchanged. Show that this observation completes the proof.