let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a being Function of Y,BOOLEAN
for PA being a_partition of Y holds All ((I_el Y),PA,G) = I_el Y
let G be Subset of (PARTITIONS Y); for a being Function of Y,BOOLEAN
for PA being a_partition of Y holds All ((I_el Y),PA,G) = I_el Y
let a be Function of Y,BOOLEAN; for PA being a_partition of Y holds All ((I_el Y),PA,G) = I_el Y
let PA be a_partition of Y; All ((I_el Y),PA,G) = I_el Y
for z being Element of Y holds (All ((I_el Y),PA,G)) . z = TRUE
hence
All ((I_el Y),PA,G) = I_el Y
by BVFUNC_1:def 11; verum