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

theorem
  (((a 'imp' b) '&' a) '&' c) 'imp' b = I_el(Y)
proof
  thus (((a 'imp' b) '&' a) '&' c) 'imp' b =((('not' a 'or' b) '&' a) '&' c)
  'imp' b by BVFUNC_4:8
    .=((('not' a '&' a) 'or' (b '&' a)) '&' c) 'imp' b by BVFUNC_1:12
    .=((O_el(Y) 'or' (b '&' a)) '&' c) 'imp' b by BVFUNC_4:5
    .=((b '&' a) '&' c) 'imp' b by BVFUNC_1:9
    .='not' ((b '&' a) '&' c) 'or' b by BVFUNC_4:8
    .=('not' (b '&' a) 'or' 'not' c) 'or' b by BVFUNC_1:14
    .=(('not' b 'or' 'not' a) 'or' 'not' c) 'or' b by BVFUNC_1:14
    .=(b 'or' ('not' b 'or' 'not' a)) 'or' 'not' c by BVFUNC_1:8
    .=((b 'or' 'not' b) 'or' 'not' a) 'or' 'not' c by BVFUNC_1:8
    .=(I_el(Y) 'or' 'not' a) 'or' 'not' c by BVFUNC_4:6
    .=I_el(Y) 'or' 'not' c by BVFUNC_1:10
    .=I_el(Y)by BVFUNC_1:10;
end;
