let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds a '<' B_SUP a,PA
let a be Element of Funcs 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
A1: for y being Element of Y holds (a 'imp' (B_SUP a,PA)) . y = (I_el Y) . y

proof

let y be Element of Y; :: 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 Def20;
then (B_SUP a,PA) . y = (I_el Y) . y by Def14;
then ('not' (a . y)) 'or' ((B_SUP a,PA) . y) = (('not' a) . y) 'or' ((I_el Y) . y) by MARGREL1:def 20
.= (('not' a) 'or' (I_el Y)) . y by Def7
.= (I_el Y) . y by Th13 ;
hence (a 'imp' (B_SUP a,PA)) . y = (I_el Y) . y by Def11; :: thesis: verum

end;

suppose A2: 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

y in EqClass y,PA by EQREL_1:def 8;
then a . y <> TRUE by A2;
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 Def14
.= ((I_el Y) 'or' (B_SUP a,PA)) . y by Def7
.= (I_el Y) . y by Th13 ;
hence (a 'imp' (B_SUP a,PA)) . y = (I_el Y) . y by Def11; :: thesis: verum

end;

end;

end;

consider k3 being Function such that
A3: ( a 'imp' (B_SUP a,PA) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A4: ( 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, A3, A4;
then a 'imp' (B_SUP a,PA) = I_el Y by A3, A4, FUNCT_1:9;
hence a '<' B_SUP a,PA by Th19; :: thesis: verum