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 All (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 All (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 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
A1: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
(O_el Y) . z = FALSE by BVFUNC_1:def 13;
hence (All (O_el Y),PA,G) . z = FALSE by A1, BVFUNC_1:def 19; :: thesis: verum

end;

hence All (O_el Y),PA,G = O_el Y by BVFUNC_1:def 13; :: thesis: verum