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 Ex ((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 Ex ((I_el Y),PA,G) = I_el Y
let a be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds Ex ((I_el Y),PA,G) = I_el Y
let PA be a_partition of Y; :: thesis: Ex ((I_el Y),PA,G) = I_el Y
for z being Element of Y holds (Ex ((I_el Y),PA,G)) . z = TRUE

proof

let z be Element of Y; :: thesis: (Ex ((I_el Y),PA,G)) . z = TRUE
( z in EqClass (z,(CompF (PA,G))) & (I_el Y) . z = TRUE ) by BVFUNC_1:def 11, EQREL_1:def 6;
hence (Ex ((I_el Y),PA,G)) . z = TRUE by BVFUNC_1:def 17; :: thesis: verum

end;

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