let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'or' c '<' a 'or' c
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a '&' b) 'or' c '<' a 'or' c
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((a '&' b) 'or' c) . z = TRUE or (a 'or' c) . z = TRUE )
A1: ((a '&' b) 'or' c) . z = ((a '&' b) . z) 'or' (c . z) by BVFUNC_1:def 4
.= ((a . z) '&' (b . z)) 'or' (c . z) by MARGREL1:def 20 ;
assume A2: ((a '&' b) 'or' 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;
((a . z) '&' (b . z)) 'or' (c . z) = ((c . z) 'or' (a . z)) '&' ((c . z) 'or' (b . z)) by XBOOLEAN:9
.= FALSE by A3 ;
hence contradiction by A2, A1; :: thesis: verum

end;

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