let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (a 'imp' ('not' b)) '<' 'not' a
let a, b be Function of Y,BOOLEAN; :: thesis: (a 'imp' b) '&' (a 'imp' ('not' b)) '<' 'not' a
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((a 'imp' b) '&' (a 'imp' ('not' b))) . z = TRUE or ('not' a) . z = TRUE )
A1: ((a 'imp' b) '&' (a 'imp' ('not' b))) . z = ((a 'imp' b) . z) '&' ((a 'imp' ('not' b)) . z) by MARGREL1:def 20
.= ((('not' a) 'or' b) . z) '&' ((a 'imp' ('not' b)) . z) by BVFUNC_4:8
.= ((('not' a) 'or' b) . z) '&' ((('not' a) 'or' ('not' b)) . z) by BVFUNC_4:8
.= ((('not' a) . z) 'or' (b . z)) '&' ((('not' a) 'or' ('not' b)) . z) by BVFUNC_1:def 4
.= ((('not' a) . z) 'or' (b . z)) '&' ((('not' a) . z) 'or' (('not' b) . z)) by BVFUNC_1:def 4 ;
assume A2: ((a 'imp' b) '&' (a 'imp' ('not' b))) . z = TRUE ; :: thesis: ('not' a) . z = TRUE

now :: thesis: not ('not' a) . z <> TRUE

assume ('not' a) . z <> TRUE ; :: thesis: contradiction
then ('not' a) . z = FALSE by XBOOLEAN:def 3;
then ((('not' a) . z) 'or' (b . z)) '&' ((('not' a) . z) 'or' (('not' b) . z)) = (b . z) '&' ('not' (b . z)) by MARGREL1:def 19
.= FALSE by XBOOLEAN:138 ;
hence contradiction by A2, A1; :: thesis: verum

end;

hence ('not' a) . z = TRUE ; :: thesis: verum