let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN st a = I_el Y holds
(a 'imp' b) 'imp' b = I_el Y
let a, b be Function of Y,BOOLEAN; :: thesis: ( a = I_el Y implies (a 'imp' b) 'imp' b = I_el Y )
assume A1: a = I_el Y ; :: thesis: (a 'imp' b) 'imp' b = I_el Y
for x being Element of Y holds ((a 'imp' b) 'imp' b) . x = TRUE

proof

let x be Element of Y; :: thesis: ((a 'imp' b) 'imp' b) . x = TRUE

A2: now :: thesis: ( ( b . x = TRUE & ('not' (b . x)) 'or' (b . x) = TRUE ) or ( b . x = FALSE & ('not' (b . x)) 'or' (b . x) = TRUE ) )

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

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

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

end;

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

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

end;

end;

end;

A3: a . x = TRUE by A1, BVFUNC_1:def 11;
((a 'imp' b) 'imp' b) . x = ('not' ((a 'imp' b) . x)) 'or' (b . x) by BVFUNC_1:def 8
.= ('not' (('not' (a . x)) 'or' (b . x))) 'or' (b . x) by BVFUNC_1:def 8
.= ((b . x) 'or' TRUE) '&' TRUE by A3, A2, XBOOLEAN:9
.= (b . x) 'or' TRUE by MARGREL1:14
.= TRUE by BINARITH:10 ;
hence ((a 'imp' b) 'imp' b) . x = TRUE ; :: thesis: verum

end;

hence (a 'imp' b) 'imp' b = I_el Y by BVFUNC_1:def 11; :: thesis: verum