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' (All (b,PA,G)) '<' All ((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' (All (b,PA,G)) '<' All ((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' (All (b,PA,G)) '<' All ((a 'imp' b),PA,G)
let PA be a_partition of Y; (Ex (a,PA,G)) 'imp' (All (b,PA,G)) '<' All ((a 'imp' b),PA,G)
let z be Element of Y; BVFUNC_1:def 12 ( not ((Ex (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE or (All ((a 'imp' b),PA,G)) . z = TRUE )
assume
((Ex (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE
; (All ((a 'imp' b),PA,G)) . z = TRUE
then A1:
('not' ((Ex (a,PA,G)) . z)) 'or' ((All (b,PA,G)) . z) = TRUE
by BVFUNC_1:def 8;
per cases
( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE or ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) or ( ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) )
;
suppose A4:
( ex
x being
Element of
Y st
(
x in EqClass (
z,
(CompF (PA,G))) &
a . x = TRUE ) & ex
x being
Element of
Y st
(
x in EqClass (
z,
(CompF (PA,G))) & not
b . x = TRUE ) )
;
(All ((a 'imp' b),PA,G)) . z = TRUE then
(B_SUP (a,(CompF (PA,G)))) . z = TRUE
by BVFUNC_1:def 17;
then
(Ex (a,PA,G)) . z = TRUE
by BVFUNC_2:def 10;
then A5:
'not' ((Ex (a,PA,G)) . z) = FALSE
by MARGREL1:11;
(B_INF (b,(CompF (PA,G)))) . z = FALSE
by A4, BVFUNC_1:def 16;
then
(All (b,PA,G)) . z = FALSE
by BVFUNC_2:def 9;
hence
(All ((a 'imp' b),PA,G)) . z = TRUE
by A1, A5;
verum end; end;