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 a '<' Ex (a,PA,G)

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

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

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

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

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

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

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

hence (Ex (a,PA,G)) . z = TRUE by A1, BVFUNC_1:def 17; :: thesis: verum

for a being Function of Y,BOOLEAN

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

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

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

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

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

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

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

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

hence (Ex (a,PA,G)) . z = TRUE by A1, BVFUNC_1:def 17; :: thesis: verum