let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a being Element of Funcs (Y,BOOLEAN)
for PA being a_partition of Y holds Ex ((O_el Y),PA,G) = O_el Y
let G be Subset of (PARTITIONS Y); :: thesis: for a being Element of Funcs (Y,BOOLEAN)
for PA being a_partition of Y holds Ex ((O_el Y),PA,G) = O_el Y
let a be Element of Funcs (Y,BOOLEAN); :: thesis: for PA being a_partition of Y holds Ex ((O_el Y),PA,G) = O_el Y
let PA be a_partition of Y; :: thesis: Ex ((O_el Y),PA,G) = O_el Y
for z being Element of Y holds (Ex ((O_el Y),PA,G)) . z = FALSE

proof

let z be Element of Y; :: thesis: (Ex ((O_el Y),PA,G)) . z = FALSE
for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(O_el Y) . x <> TRUE by BVFUNC_1:def 10;
hence (Ex ((O_el Y),PA,G)) . z = FALSE by BVFUNC_1:def 17; :: thesis: verum

end;

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