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 ((O_el Y),PA,G) = O_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 ((O_el Y),PA,G) = O_el Y
let a be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All ((O_el Y),PA,G) = O_el Y
let PA be a_partition of Y; :: thesis: All ((O_el Y),PA,G) = O_el Y
for z being Element of Y holds (All ((O_el Y),PA,G)) . z = FALSE

proof

let z be Element of Y; :: thesis: (All ((O_el Y),PA,G)) . z = FALSE
( z in EqClass (z,(CompF (PA,G))) & (O_el Y) . z = FALSE ) by BVFUNC_1:def 10, EQREL_1:def 6;
hence (All ((O_el Y),PA,G)) . z = FALSE by BVFUNC_1:def 16; :: thesis: verum

end;

hence All ((O_el Y),PA,G) = O_el Y by BVFUNC_1:def 10; :: thesis: verum