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

proof

let y be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a 'imp' (B_SUP (a,PA))) . y = (I_el Y) . y

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

suppose ex x being Element of Y st
( x in EqClass (y,PA) & a . x = TRUE ) ; :: thesis: (a 'imp' (B_SUP (a,PA))) . y = (I_el Y) . y

then (B_SUP (a,PA)) . y = TRUE by Def17;
then (B_SUP (a,PA)) . y = (I_el Y) . y by Def11;
then ('not' (a . y)) 'or' ((B_SUP (a,PA)) . y) = (('not' a) . y) 'or' ((I_el Y) . y) by MARGREL1:def 19
.= (('not' a) 'or' (I_el Y)) . y by Def4
.= (I_el Y) . y by Th9 ;
hence (a 'imp' (B_SUP (a,PA))) . y = (I_el Y) . y by Def8; :: thesis: verum

end;

suppose A1: for x being Element of Y holds
( not x in EqClass (y,PA) or not a . x = TRUE ) ; :: thesis: (a 'imp' (B_SUP (a,PA))) . y = (I_el Y) . y

a . y <> TRUE by A1, EQREL_1:def 6;
then a . y = FALSE by XBOOLEAN:def 3;
then ('not' (a . y)) 'or' ((B_SUP (a,PA)) . y) = ((I_el Y) . y) 'or' ((B_SUP (a,PA)) . y) by Def11
.= ((I_el Y) 'or' (B_SUP (a,PA))) . y by Def4
.= (I_el Y) . y by Th9 ;
hence (a 'imp' (B_SUP (a,PA))) . y = (I_el Y) . y by Def8; :: thesis: verum

end;

end;

end;

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