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 ((I_el Y),PA,G) = I_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 ((I_el Y),PA,G) = I_el Y

let a be Function of 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

for a being Function of 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 Function of Y,BOOLEAN

for PA being a_partition of Y holds All ((I_el Y),PA,G) = I_el Y

let a be Function of 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

hence
All ((I_el Y),PA,G) = I_el Y
by BVFUNC_1:def 11; :: thesis: verum
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 11;

hence (All ((I_el Y),PA,G)) . z = TRUE by BVFUNC_1:def 16; :: thesis: verum

end;for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

(I_el Y) . x = TRUE by BVFUNC_1:def 11;

hence (All ((I_el Y),PA,G)) . z = TRUE by BVFUNC_1:def 16; :: thesis: verum