reserve Y for non empty set,
  a,b,c,d for Function of Y,BOOLEAN;

theorem
  a '&' (a 'imp' b) '&' (b 'imp' c) '<' c
proof
  (a '&' (a 'imp' b) '&' (b 'imp' c)) 'imp' c ='not' (a '&' (a 'imp' b)
  '&' (b 'imp' c)) 'or' c by BVFUNC_4:8
    .='not' (a '&' b '&' (b 'imp' c)) 'or' c by BVFUNC_6:56
    .='not' (a '&' (b '&' (b 'imp' c))) 'or' c by BVFUNC_1:4
    .='not' (a '&' (b '&' c)) 'or' c by BVFUNC_6:56
    .='not' ((a '&' b) '&' c) 'or' c by BVFUNC_1:4
    .=('not' (a '&' b) 'or' 'not' c) 'or' c by BVFUNC_1:14
    .='not' (a '&' b) 'or' ('not' c 'or' c) by BVFUNC_1:8
    .='not' (a '&' b) 'or' I_el(Y) by BVFUNC_4:6
    .= I_el(Y) by BVFUNC_1:10;
  hence thesis by BVFUNC_1:16;
end;
