reserve Y for non empty set;

theorem
  for a being Function of Y,BOOLEAN holds ('not' a 'imp' a) 'imp'
  a = I_el(Y)
proof
  let a be Function of Y,BOOLEAN;
    let x be Element of Y;
A1: 'not' ('not' ('not' a).x 'or' a.x) = 'not' (a.x 'or' a.x) by
MARGREL1:def 19
      .= 'not' a.x;
A2: ('not' a 'imp' a 'imp' a).x =('not' ('not' a 'imp' a).x) 'or' a.x by
BVFUNC_1:def 8
      .='not' a.x 'or' a.x by A1,BVFUNC_1:def 8;
A3: (I_el Y).x=TRUE by BVFUNC_1:def 11;
    now
      per cases by XBOOLEAN:def 3;
      case
        a.x=TRUE;
        hence thesis by A2,A3,BINARITH:10;
      end;
      case
        a.x=FALSE;
        then ('not' a 'imp' a 'imp' a).x =TRUE 'or' FALSE by A2,MARGREL1:11
          .=TRUE by BINARITH:10;
        hence thesis by BVFUNC_1:def 11;
      end;
    end;
    hence thesis;
end;
