:: deftheorem Def3 defines $$ SETWISEO:def 3 :
for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st ( B <> {} or F is having_a_unity ) & F is commutative & F is associative holds
for b₆ being Element of Y holds
( b₆ = F $$ (B,f) iff ex G being Function of (Fin X),Y st
( b₆ = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B \ B9 holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) ) );