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

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

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

end;

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