let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds (a 'imp' ('not' a)) 'imp' ('not' a) = I_el Y
let a be Function of Y,BOOLEAN; :: thesis: (a 'imp' ('not' a)) 'imp' ('not' a) = I_el Y
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a 'imp' ('not' a)) 'imp' ('not' a)) . x = (I_el Y) . x

A1: now :: thesis: ( ( a . x = TRUE & ('not' (a . x)) 'or' (a . x) = TRUE ) or ( a . x = FALSE & ('not' (a . x)) 'or' (a . x) = TRUE ) )

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 A1, MARGREL1:def 19 ;
hence ((a 'imp' ('not' a)) 'imp' ('not' a)) . x = (I_el Y) . x by BVFUNC_1:def 11; :: thesis: verum