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 (a,PA,G) '<' a

let G be Subset of (PARTITIONS Y); :: thesis: for a being Function of Y,BOOLEAN

for PA being a_partition of Y holds All (a,PA,G) '<' a

let a be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All (a,PA,G) '<' a

let PA be a_partition of Y; :: thesis: All (a,PA,G) '<' a

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All (a,PA,G)) . z = TRUE or a . z = TRUE )

A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

assume (All (a,PA,G)) . z = TRUE ; :: thesis: a . z = TRUE

hence a . z = TRUE by A1, BVFUNC_1:def 16; :: thesis: verum

for a being Function of Y,BOOLEAN

for PA being a_partition of Y holds All (a,PA,G) '<' a

let G be Subset of (PARTITIONS Y); :: thesis: for a being Function of Y,BOOLEAN

for PA being a_partition of Y holds All (a,PA,G) '<' a

let a be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All (a,PA,G) '<' a

let PA be a_partition of Y; :: thesis: All (a,PA,G) '<' a

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All (a,PA,G)) . z = TRUE or a . z = TRUE )

A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

assume (All (a,PA,G)) . z = TRUE ; :: thesis: a . z = TRUE

hence a . z = TRUE by A1, BVFUNC_1:def 16; :: thesis: verum