let Y be non empty set ; :: thesis: 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); :: thesis: 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; :: thesis: for PA being a_partition of Y holds All ((I_el Y),PA,G) = I_el Y
let PA be a_partition of Y; :: thesis: All ((I_el Y),PA,G) = I_el Y
for z being Element of Y holds (All ((I_el Y),PA,G)) . z = TRUE

proof

let z be Element of Y; :: thesis: (All ((I_el Y),PA,G)) . z = TRUE
for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(I_el Y) . x = TRUE by BVFUNC_1:def 11;
hence (All ((I_el Y),PA,G)) . z = TRUE by BVFUNC_1:def 16; :: thesis: verum

end;

hence All ((I_el Y),PA,G) = I_el Y by BVFUNC_1:def 11; :: thesis: verum