let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds (('not' a) 'imp' a) 'imp' a = I_el Y
let a be Function of Y,BOOLEAN; :: thesis: (('not' a) 'imp' a) 'imp' a = I_el Y
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x
A1: 'not' (('not' (('not' a) . x)) 'or' (a . x)) = 'not' ((a . x) 'or' (a . x)) by MARGREL1:def 19
.= 'not' (a . x) ;
A2: ((('not' a) 'imp' a) 'imp' a) . x = ('not' ((('not' a) 'imp' a) . x)) 'or' (a . x) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' (a . x) by A1, BVFUNC_1:def 8 ;
A3: (I_el Y) . x = TRUE by BVFUNC_1:def 11;

now :: thesis: ( ( a . x = TRUE & ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x ) or ( a . x = FALSE & ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x ) )

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 A2, A3, BINARITH:10; :: 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 A2, MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ((('not' a) 'imp' a) 'imp' a) . x = (I_el Y) . x by BVFUNC_1:def 11; :: thesis: verum

end;

end;

end;

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