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
consider k3 being Function such that
A1: (('not' a) 'imp' a) 'imp' 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 ((('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
A5: 'not' (('not' (('not' a) . x)) 'or' (a . x)) = 'not' ((a . x) 'or' (a . x)) by MARGREL1:def 20
.= 'not' (a . x) ;
A6: ((('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 A5, BVFUNC_1:def 11 ;
A7: (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 A6, A7, 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 A6, 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;

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