let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN
for PA being a_partition of Y holds B_INF (a,PA) '<' a
let a be Function of 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
(B_INF (a,PA)) 'imp' a = I_el Y

proof

let y be Element of Y; :: according to FUNCT_2:def 8 :: 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 A1: 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

A2: a . y = TRUE by A1, EQREL_1:def 6;
'not' ((B_INF (a,PA)) . y) = ('not' (B_INF (a,PA))) . y by MARGREL1:def 19;
then ('not' ((B_INF (a,PA)) . y)) 'or' (a . y) = (('not' (B_INF (a,PA))) . y) 'or' ((I_el Y) . y) by A2, Def11
.= (('not' (B_INF (a,PA))) 'or' (I_el Y)) . y by Def4
.= (I_el Y) . y by Th9 ;
hence ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y by Def8; :: 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 Def16;
then ('not' ((B_INF (a,PA)) . y)) 'or' (a . y) = ((I_el Y) . y) 'or' (a . y) by Def11
.= ((I_el Y) 'or' a) . y by Def4
.= (I_el Y) . y by Th9 ;
hence ((B_INF (a,PA)) 'imp' a) . y = (I_el Y) . y by Def8; :: thesis: verum

end;

end;

end;

hence B_INF (a,PA) '<' a by Th15; :: thesis: verum