reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) '&' c)
  'imp' ((a '&' c) 'or' (b '&' c))=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  for x being Element of Y holds (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or'
  (b '&' c))).x=TRUE
  proof
    let x be Element of Y;
    (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))).x ='not' ((a
    'or' b) '&' c).x 'or' ((a '&' c) 'or' (b '&' c)).x by BVFUNC_1:def 8
      .='not'( ((a 'or' b)).x '&' c.x) 'or' ((a '&' c) 'or' (b '&' c)).x
    by MARGREL1:def 20
      .='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a '&' c) 'or' (b '&' c))
    .x by BVFUNC_1:def 4
      .='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a '&' c).x 'or' (b '&' c
    ).x) by BVFUNC_1:def 4
      .='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a.x '&' c.x) 'or' (b
    '&' c).x) by MARGREL1:def 20
      .='not'( c.x '&' (a.x 'or' b.x)) 'or' ((a.x '&' c.x) 'or' ((
    b).x '&' c.x)) by MARGREL1:def 20
      .=((a.x '&' c.x) 'or' (b.x '&' c.x)) 'or' ('not'( a.x '&' (c
    ).x) '&' 'not'( b.x '&' c.x)) by XBOOLEAN:8
      .=(((b.x '&' c.x) 'or' (a.x '&' c.x)) 'or' 'not'( a.x '&' (c
).x)) '&' (((a.x '&' c.x) 'or' (b.x '&' c.x)) 'or' 'not'( b.x '&' (c)
    .x)) by XBOOLEAN:9
      .=((b.x '&' c.x) 'or' ((a.x '&' c.x) 'or' 'not'( a.x '&' (c)
.x))) '&' ((a.x '&' c.x) 'or' ((b.x '&' c.x) 'or' 'not'( b.x '&' c.
    x)))
      .=((b.x '&' c.x) 'or' TRUE) '&' ((a.x '&' c.x) 'or' ((b.x
    '&' c.x) 'or' 'not'( b.x '&' c.x))) by XBOOLEAN:102
      .=TRUE '&' ((a.x '&' c.x) 'or' TRUE) by XBOOLEAN:102
      .=TRUE;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
