let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN holds a 'imp' ('not' a) = 'not' a
let a be Element of Funcs Y,BOOLEAN ; :: thesis: a 'imp' ('not' a) = 'not' a
consider k3 being Function such that
A1: a 'imp' ('not' a) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not' a = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds (a 'imp' ('not' a)) . x = ('not' a) . x

proof

let x be Element of Y; :: thesis: (a 'imp' ('not' a)) . x = ('not' a) . x
(a 'imp' ('not' a)) . x = ('not' (a . x)) 'or' (('not' a) . x) by BVFUNC_1:def 11
.= (('not' a) . x) 'or' (('not' a) . x) by MARGREL1:def 20
.= ('not' a) . x ;
hence (a 'imp' ('not' a)) . x = ('not' a) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence a 'imp' ('not' a) = 'not' a by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum