let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let G be Subset of (PARTITIONS Y); for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let a, b be Function of Y,BOOLEAN; for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let PA be a_partition of Y; (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let z be Element of Y; BVFUNC_1:def 12 ( not ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE or (Ex ((a 'imp' b),PA,G)) . z = TRUE )
A1:
z in EqClass (z,(CompF (PA,G)))
by EQREL_1:def 6;
assume
((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE
; (Ex ((a 'imp' b),PA,G)) . z = TRUE
then A2:
('not' ((Ex (a,PA,G)) . z)) 'or' ((Ex (b,PA,G)) . z) = TRUE
by BVFUNC_1:def 8;
A3:
( 'not' ((Ex (a,PA,G)) . z) = TRUE or 'not' ((Ex (a,PA,G)) . z) = FALSE )
by XBOOLEAN:def 3;
hence
(Ex ((a 'imp' b),PA,G)) . z = TRUE
; verum