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

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

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

end;

hence (b 'or' c) . z = TRUE ; :: thesis: verum