let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN)
for PA being a_partition of Y holds B_INF (a,PA) '<' a
let a be Element of Funcs (Y,BOOLEAN); :: thesis: for PA being a_partition of Y holds B_INF (a,PA) '<' a
let PA be a_partition of Y; :: thesis: B_INF (a,PA) '<' a
consider k3 being Function such that
A1: (B_INF (a,PA)) 'imp' a = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: I_el Y = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for y being Element of Y holds ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y

proof

let y be Element of Y; :: thesis: ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y

per cases ( for x being Element of Y st x in EqClass (y,PA) holds
a . x = TRUE or ex x being Element of Y st
( x in EqClass (y,PA) & not a . x = TRUE ) ) ;

suppose A5: for x being Element of Y st x in EqClass (y,PA) holds
a . x = TRUE ; :: thesis: ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y

y in EqClass (y,PA) by EQREL_1:def 8;
then A6: a . y = TRUE by A5;
'not' ((B_INF (a,PA)) . y) = ('not' (B_INF (a,PA))) . y by MARGREL1:def 20;
then ('not' ((B_INF (a,PA)) . y)) 'or' (a . y) = (('not' (B_INF (a,PA))) . y) 'or' ((I_el Y) . y) by A6, Def14
.= (('not' (B_INF (a,PA))) 'or' (I_el Y)) . y by Def7
.= (I_el Y) . y by Th13 ;
hence ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y by Def11; :: thesis: verum

end;

suppose ex x being Element of Y st
( x in EqClass (y,PA) & not a . x = TRUE ) ; :: thesis: ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y

then (B_INF (a,PA)) . y = FALSE by Def19;
then ('not' ((B_INF (a,PA)) . y)) 'or' (a . y) = ((I_el Y) . y) 'or' (a . y) by Def14
.= ((I_el Y) 'or' a) . y by Def7
.= (I_el Y) . y by Th13 ;
hence ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y by Def11; :: thesis: verum

end;

end;

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
then k3 = k4 by A2, A4, FUNCT_1:9;
hence B_INF (a,PA) '<' a by A1, A3, Th19; :: thesis: verum