reserve Y for non empty set;

theorem
  for a being Function of Y,BOOLEAN holds (a 'imp' 'not' a) 'imp'
  'not' a=I_el(Y)
proof
  let a 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;
    ((a 'imp' 'not' a) 'imp' 'not' a).x ='not' (a 'imp' 'not' a).x 'or' (
    'not' a).x by BVFUNC_1:def 8
      .='not' ('not' a.x 'or' ('not' a).x) 'or' ('not' a).x by BVFUNC_1:def 8
      .=(a.x '&' 'not' 'not' a.x) 'or' ('not' a).x by MARGREL1:def 19
      .=TRUE by A1,MARGREL1:def 19;
    hence thesis by BVFUNC_1:def 11;
end;
