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 (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 Ex (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 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
A1: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
(I_el Y) . z = TRUE by BVFUNC_1:def 14;
hence (Ex (I_el Y),PA,G) . z = TRUE by A1, BVFUNC_1:def 20; :: thesis: verum

end;

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