let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN holds a '&' ('not' a) = O_el Y
let a be Element of Funcs Y,BOOLEAN ; :: thesis: a '&' ('not' a) = O_el Y
A1: for x being Element of Y holds (a '&' ('not' a)) . x = (O_el Y) . x

proof

let x be Element of Y; :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x
A2: (a '&' ('not' a)) . x = (a . x) '&' (('not' a) . x) by MARGREL1:def 21
.= (a . x) '&' ('not' (a . x)) by MARGREL1:def 20 ;

now

per cases ( a . x = TRUE or a . x = FALSE ) by XBOOLEAN:def 3;

case a . x = TRUE ; :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x

hence (a '&' ('not' a)) . x = (O_el Y) . x by A2, BVFUNC_1:def 13; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x

hence (a '&' ('not' a)) . x = (O_el Y) . x by A2, BVFUNC_1:def 13; :: thesis: verum

end;

end;

end;

hence (a '&' ('not' a)) . x = (O_el Y) . x ; :: thesis: verum

end;

consider k3 being Function such that
A3: ( a '&' ('not' a) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A4: ( O_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;
hence a '&' ('not' a) = O_el Y by A3, A4, FUNCT_1:9; :: thesis: verum