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

proof

let x be Element of Y; :: thesis: ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x
A2: 'not' (('not' (('not' a) . x)) 'or' (a . x)) = 'not' ((a . x) 'or' (a . x)) by MARGREL1:def 20
.= 'not' (a . x) ;
A3: ((('not' a) 'imp' a) 'imp' a) . x = ('not' ((('not' a) 'imp' a) . x)) 'or' (a . x) by BVFUNC_1:def 11
.= ('not' (a . x)) 'or' (a . x) by A2, BVFUNC_1:def 11 ;
A4: (I_el Y) . x = TRUE by BVFUNC_1:def 14;

now

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

case a . x = TRUE ; :: thesis: ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x

hence ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x by A3, A4, BINARITH:19; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x

then ((('not' a) 'imp' a) 'imp' a) . x = TRUE 'or' FALSE by A3, MARGREL1:41
.= TRUE by BINARITH:19 ;
hence ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x by BVFUNC_1:def 14; :: thesis: verum

end;

end;

end;

hence ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x ; :: thesis: verum

end;

consider k3 being Function such that
A5: ( (('not' a) 'imp' a) 'imp' a = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A6: ( I_el Y = 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, A5, A6;
hence (('not' a) 'imp' a) 'imp' a = I_el Y by A5, A6, FUNCT_1:9; :: thesis: verum