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

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

assume (('not' a) 'or' ('not' b)) . z <> TRUE ; :: thesis: contradiction
then (('not' a) 'or' ('not' b)) . z = FALSE by XBOOLEAN:def 3;
then A3: (('not' a) . z) 'or' (('not' b) . z) = FALSE by BVFUNC_1:def 4;
( ('not' b) . z = TRUE or ('not' b) . z = FALSE ) by XBOOLEAN:def 3;
then ((('not' a) . z) 'or' (c . z)) '&' ((('not' b) . z) 'or' (('not' c) . z)) = (c . z) '&' ('not' (c . z)) by A3, MARGREL1:def 19
.= FALSE by XBOOLEAN:138 ;
hence contradiction by A2, A1; :: thesis: verum

end;

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