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

now :: thesis: ( ((a '&' c) 'imp' (b '&' d)) . z <> TRUE implies ((a '&' c) 'imp' (b '&' d)) . z = TRUE )

A3: ( ('not' c) . z = TRUE or ('not' c) . z = FALSE ) by XBOOLEAN:def 3;
A4: ( (('not' a) . z) 'or' (('not' c) . z) = TRUE or (('not' a) . z) 'or' (('not' c) . z) = FALSE ) by XBOOLEAN:def 3;
A5: ( (b . z) '&' (d . z) = TRUE or (b . z) '&' (d . z) = FALSE ) by XBOOLEAN:def 3;
A6: ((a '&' c) 'imp' (b '&' d)) . z = (('not' (a '&' c)) 'or' (b '&' d)) . z by BVFUNC_4:8
.= (('not' (a '&' c)) . z) 'or' ((b '&' d) . z) by BVFUNC_1:def 4
.= ((('not' a) 'or' ('not' c)) . z) 'or' ((b '&' d) . z) by BVFUNC_1:14
.= ((('not' a) . z) 'or' (('not' c) . z)) 'or' ((b '&' d) . z) by BVFUNC_1:def 4
.= ((('not' a) . z) 'or' (('not' c) . z)) 'or' ((b . z) '&' (d . z)) by MARGREL1:def 20 ;
assume A7: ((a '&' c) 'imp' (b '&' d)) . z <> TRUE ; :: thesis: ((a '&' c) 'imp' (b '&' d)) . z = TRUE

now :: thesis: ( ( b . z = FALSE & ((a '&' c) 'imp' (b '&' d)) . z = TRUE ) or ( d . z = FALSE & ((a '&' c) 'imp' (b '&' d)) . z = TRUE ) )

per cases ( b . z = FALSE or d . z = FALSE ) by A7, A6, A5, MARGREL1:12;

case b . z = FALSE ; :: thesis: ((a '&' c) 'imp' (b '&' d)) . z = TRUE

thus ((a '&' c) 'imp' (b '&' d)) . z = TRUE by A2, A1, A6, A4, A3; :: thesis: verum

end;

case d . z = FALSE ; :: thesis: ((a '&' c) 'imp' (b '&' d)) . z = TRUE

thus ((a '&' c) 'imp' (b '&' d)) . z = TRUE by A2, A1, A6, A4, A3; :: thesis: verum

end;

end;

end;

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

end;

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