Claim:

=

Proof:

means to take the union of 1( r , a )'s for all states r that are reachable from some state in 1*( q , w ) through zero or more arcs. Since 1*( q , w ) is a -closure, any state reachable from any of its states through zero or more arcs is already in 1*( q , w ). Hence to find all we need is to take the union of 1( r , a )'s for all r in 1*( q , w ) , which is .
Hence =