reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds (a 'imp' (b 'imp' c)
  ) 'imp' ((a '&' b) 'imp' c)=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  for x being Element of Y holds ((a 'imp' (b 'imp' c)) 'imp' ((a '&' b)
  'imp' c)).x=TRUE
  proof
    let x be Element of Y;
    ((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)).x ='not' (a 'imp' (b
    'imp' c)).x 'or' ((a '&' b) 'imp' c).x by BVFUNC_1:def 8
      .='not'( 'not' a.x 'or' (b 'imp' c).x) 'or' ((a '&' b) 'imp' c).x by
BVFUNC_1:def 8
      .='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ((a '&' b)
    'imp' c).x by BVFUNC_1:def 8
      .='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ('not' (a '&'
    b).x 'or' c.x) by BVFUNC_1:def 8
      .='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' (('not' a.x
    'or' 'not' b.x) 'or' c.x) by MARGREL1:def 20
      .=TRUE by XBOOLEAN:102;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
