Let ${\displaystyle M\models }$  ZFC be transitive, ${\displaystyle (\mathbb {P} ,<)}$  a forcing notion in ${\displaystyle M}$ , and suppose ${\displaystyle G\subseteq \mathbb {P} }$  is generic over ${\displaystyle M}$ .

Then for any ${\displaystyle \mathbb {P} }$ -name ${\displaystyle \tau }$  in ${\displaystyle M}$ , we say that ${\displaystyle \eta }$  is a nice name for a subset of ${\displaystyle \tau }$  if ${\displaystyle \eta }$  is a ${\displaystyle \mathbb {P} }$ -name satisfying the following properties:

(1) ${\displaystyle \operatorname {dom} (\eta )\subseteq \operatorname {dom} (\tau )}$ 

(2) For all ${\displaystyle \mathbb {P} }$ -names ${\displaystyle \sigma \in M}$ , ${\displaystyle \{p\in \mathbb {P} |\langle \sigma ,p\rangle \in \eta \}}$  forms an antichain.

(3) (Natural addition): If ${\displaystyle \langle \sigma ,p\rangle \in \eta }$ , then there exists ${\displaystyle q\geq p}$  in ${\displaystyle \mathbb {P} }$  such that ${\displaystyle \langle \sigma ,q\rangle \in \tau }$ .