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To express in quantifier notation that Al and Sue have exactly one child we write:
$\displaystyle \exists !y$[ F( A,y) & M( S,y)] ,which means there is a unique child (y) ,lets say ,of whom Al and Sue are the father and mother respectively:
Now that formula is equivalent to the following formula:
$\displaystyle \exists y$[ F( A,y) & M( S,y)]& $\displaystyle \forall y\forall z${[F( A,y) & M( S,Y)] & [ F( A,z) & M( S,z)]------> y=z}:
The 2nd formula means that Sue and Al have a child and their child is the only one (expressed by the 2nd part of the formula:$\displaystyle \forall y\forall z${[F( A,y) & M( S,Y)] & [ F( A,z) & M( S,z)]------> y=z}:
NOW if my 2nd formula is equivalent to that given by Plato ,one should be able to prove that.
I think it is a good exercise for you