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

proof

let x be Element of Y; :: thesis: (a 'eqv' (O_el Y)) . x = ('not' a) . x
(a 'eqv' (O_el Y)) . x = ((a 'imp' (O_el Y)) '&' ((O_el Y) 'imp' a)) . x by BVFUNC_4:7
.= ((('not' a) 'or' (O_el Y)) '&' ((O_el Y) 'imp' a)) . x by BVFUNC_4:8
.= ((('not' a) 'or' (O_el Y)) '&' (('not' (O_el Y)) 'or' a)) . x by BVFUNC_4:8
.= (('not' a) '&' (('not' (O_el Y)) 'or' a)) . x by BVFUNC_1:12
.= (('not' a) '&' ((I_el Y) 'or' a)) . x by BVFUNC_1:5
.= (('not' a) '&' (I_el Y)) . x by BVFUNC_1:13
.= ('not' a) . x by BVFUNC_1:9 ;
hence (a 'eqv' (O_el Y)) . x = ('not' a) . x ; :: thesis: verum

end;

consider k3 being Function such that
A2: ( a 'eqv' (O_el Y) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A3: ( 'not' a = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
( Y = dom k3 & Y = dom k4 & ( for u being set st u in Y holds
k3 . u = k4 . u ) ) by A1, A2, A3;
hence a 'eqv' (O_el Y) = 'not' a by A2, A3, FUNCT_1:9; :: thesis: verum