reserve Y for non empty set;

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