reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN holds a=I_el(Y) & b=I_el(Y)
  iff (a '&' b)=I_el(Y)
proof
  let a,b be Function of Y,BOOLEAN;
  now
    assume
A1: a '&' b=I_el(Y);
    per cases;
    suppose
      a=I_el(Y) & b=I_el(Y);
      hence a=I_el(Y) & b=I_el(Y);
    end;
    suppose
      a=I_el(Y) & b<>I_el(Y);
      hence a=I_el(Y) & b=I_el(Y) by A1,BVFUNC_1:6;
    end;
    suppose
      a<>I_el(Y) & b=I_el(Y);
      hence a=I_el(Y) & b=I_el(Y) by A1,BVFUNC_1:6;
    end;
    suppose
A2:   a<>I_el(Y) & b<>I_el(Y);
      for x being Element of Y holds a.x=TRUE
      proof
        let x be Element of Y;
        (a '&' b).x=TRUE by A1,BVFUNC_1:def 11;
        then a.x '&' b.x=TRUE by MARGREL1:def 20;
        hence thesis by MARGREL1:12;
      end;
      hence a=I_el(Y) & b=I_el(Y) by A2,BVFUNC_1:def 11;
    end;
  end;
  hence thesis;
end;
