reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds (b 'imp' c) 'imp' ((
  a 'imp' b) 'imp' (a 'imp' c))=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
    let x be Element of Y;
A1: now
      per cases by XBOOLEAN:def 3;
      case
        a.x=TRUE;
        hence ('not' a.x 'or' a.x)=TRUE by BINARITH:10;
      end;
      case
        a.x=FALSE;
        then 'not' a.x 'or' a.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' a.x 'or' a.x)=TRUE;
      end;
    end;
A2: now
      per cases by XBOOLEAN:def 3;
      case
        b.x=TRUE;
        hence ('not' b.x 'or' b.x)=TRUE by BINARITH:10;
      end;
      case
        b.x=FALSE;
        then 'not' b.x 'or' b.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' b.x 'or' b.x)=TRUE;
      end;
    end;
A3: now
      per cases by XBOOLEAN:def 3;
      case
        c.x=TRUE;
        hence ('not' c.x 'or' c.x)=TRUE by BINARITH:10;
      end;
      case
        c.x=FALSE;
        then 'not' c.x 'or' c.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' c.x 'or' c.x)=TRUE;
      end;
    end;
    ((b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))).x ='not' (b 'imp'
    c).x 'or' ((a 'imp' b) 'imp' (a 'imp' c)).x by BVFUNC_1:def 8
      .='not' (b 'imp' c).x 'or' ('not' (a 'imp' b).x 'or' (a 'imp' c).x) by
BVFUNC_1:def 8
      .='not' ('not' b.x 'or' c.x) 'or' ('not' (a 'imp' b).x 'or' (a 'imp' c
    ).x) by BVFUNC_1:def 8
      .='not' ('not' b.x 'or' c.x) 'or' ('not' ('not' a.x 'or' b.x) 'or' (a
    'imp' c).x) by BVFUNC_1:def 8
      .=(b.x '&' 'not' c.x) 'or' (('not' 'not' a.x '&' 'not' b.x) 'or' (
    'not' a.x 'or' c.x)) by BVFUNC_1:def 8
      .=(((a.x '&' 'not' b.x) 'or' ('not' a.x 'or' c.x)) 'or' 'not' c.x) '&'
    (((a.x '&' 'not' b.x) 'or' ('not' a.x 'or' c.x)) 'or' b.x) by XBOOLEAN:9
      .=((a.x '&' 'not' b.x) 'or' (('not' a.x 'or' c.x) 'or' 'not' c.x)) '&'
    (((a.x '&' 'not' b.x) 'or' ('not' a.x 'or' c.x)) 'or' b.x) by BINARITH:11
      .=((a.x '&' 'not' b.x) 'or' ('not' a.x 'or' TRUE)) '&' (((a.x '&'
    'not' b.x) 'or' ('not' a.x 'or' c.x)) 'or' b.x) by A3,BINARITH:11
      .=((a.x '&' 'not' b.x) 'or' TRUE) '&' (((a.x '&' 'not' b.x) 'or' (
    'not' a.x 'or' c.x)) 'or' b.x) by BINARITH:10
      .=TRUE '&' (((a.x '&' 'not' b.x) 'or' ('not' a.x 'or' c.x)) 'or' b.x)
    by BINARITH:10
      .=(('not' b.x '&' a.x) 'or' ('not' a.x 'or' c.x)) 'or' b.x by MARGREL1:14
      .=((('not' a.x 'or' c.x) 'or' 'not' b.x) '&' (('not' a.x 'or' c.x)
    'or' a.x)) 'or' b.x by XBOOLEAN:9
      .=((('not' a.x 'or' c.x) 'or' 'not' b.x) '&' (c.x 'or' ('not' a.x 'or'
    a.x))) 'or' b.x by BINARITH:11
      .=((('not' a.x 'or' c.x) 'or' 'not' b.x) '&' TRUE) 'or' b.x by A1,
BINARITH:10
      .=(('not' a.x 'or' c.x) 'or' 'not' b.x) 'or' b.x by MARGREL1:14
      .=('not' a.x 'or' c.x) 'or' TRUE by A2,BINARITH:11
      .=TRUE by BINARITH:10;
    hence thesis by BVFUNC_1:def 11;
end;
