let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN) holds (a 'imp' ('not' a)) 'imp' ('not' a) = I_el Y
let a be Element of Funcs (Y,BOOLEAN); :: thesis: (a 'imp' ('not' a)) 'imp' ('not' a) = I_el Y
consider k3 being Function such that
A1: (a 'imp' ('not' 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: I_el Y = 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)) 'imp' ('not' a)) . x = (I_el Y) . x

proof

let x be Element of Y; :: thesis: ((a 'imp' ('not' a)) 'imp' ('not' a)) . x = (I_el Y) . x

A5: now

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

case a . x = TRUE ; :: thesis: ('not' (a . x)) 'or' (a . x) = TRUE

hence ('not' (a . x)) 'or' (a . x) = TRUE by BINARITH:10; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: ('not' (a . x)) 'or' (a . x) = TRUE

then ('not' (a . x)) 'or' (a . x) = TRUE 'or' FALSE by MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ('not' (a . x)) 'or' (a . x) = TRUE ; :: thesis: verum

end;

end;

end;

((a 'imp' ('not' a)) 'imp' ('not' a)) . x = ('not' ((a 'imp' ('not' a)) . x)) 'or' (('not' a) . x) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (('not' a) . x))) 'or' (('not' a) . x) by BVFUNC_1:def 8
.= ((a . x) '&' ('not' ('not' (a . x)))) 'or' (('not' a) . x) by MARGREL1:def 19
.= TRUE by A5, MARGREL1:def 19 ;
hence ((a 'imp' ('not' a)) 'imp' ('not' a)) . x = (I_el Y) . x by BVFUNC_1:def 11; :: 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)) 'imp' ('not' a) = I_el Y by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum