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

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

assume (a 'imp' b) . z <> TRUE ; :: thesis: contradiction
then (a 'imp' b) . z = FALSE by XBOOLEAN:def 3;
then (('not' a) 'or' b) . z = FALSE by BVFUNC_4:8;
then A3: (('not' a) . z) 'or' (b . z) = FALSE by BVFUNC_1:def 4;
(('not' a) . z) 'or' ((b . z) '&' (c . z)) = ((('not' a) . z) 'or' (b . z)) '&' ((('not' a) . z) 'or' (c . z)) by XBOOLEAN:9
.= FALSE by A3 ;
hence contradiction by A2, A1; :: thesis: verum

end;

hence (a 'imp' b) . z = TRUE ; :: thesis: verum