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 (I_el Y),PA,G = I_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 (I_el Y),PA,G = I_el Y
let a be Element of Funcs 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 14;
hence (All (I_el Y),PA,G) . z = TRUE by BVFUNC_1:def 19; :: thesis: verum

end;

hence All (I_el Y),PA,G = I_el Y by BVFUNC_1:def 14; :: thesis: verum