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 All a,PA,G '<' a
let G be Subset of (PARTITIONS Y); :: thesis: for a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds All a,PA,G '<' a
let a be Element of Funcs 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 15 :: thesis: ( not (All a,PA,G) . z = TRUE or a . z = TRUE )
A1: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
assume (All a,PA,G) . z = TRUE ; :: thesis: a . z = TRUE
hence a . z = TRUE by A1, BVFUNC_1:def 19; :: thesis: verum