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 a '<' Ex a,PA,G
let G be Subset of (PARTITIONS Y); :: thesis: for a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds a '<' Ex a,PA,G
let a be Element of Funcs 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 15 :: thesis: ( not a . z = TRUE or (Ex a,PA,G) . z = TRUE )
A1: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
assume a . z = TRUE ; :: thesis: (Ex a,PA,G) . z = TRUE
hence (Ex a,PA,G) . z = TRUE by A1, BVFUNC_1:def 20; :: thesis: verum