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
A1: 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 A2: 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 A3: a . y = TRUE by A2;
'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 A3, 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;

consider k3 being Function such that
A4: ( (B_INF a,PA) 'imp' a = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A5: ( I_el Y = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A1, A4, A5;
then k3 = k4 by A4, A5, FUNCT_1:9;
hence B_INF a,PA '<' a by A4, A5, Th19; :: thesis: verum