The following two theorems prove the equivalence of Moore and Mealy machines in this sense.
Theorem : If 
is a Moore machine, then there is a Mealy machine 
and 
.
Proof :
We construct a Mealy machine
from the given Moore machine , where all the elements except 
are as in . is defined as
That is, the output associated with state q in the Moore machine will be associated with
the transitions going to the state q (from other state) on the same input symbol 
in the Mealy machine as shown below –