
theorem :: property (2)
  for R being non empty RelStr,
      I being Function of the carrier of R, bool the carrier of R
  st I is map-reflexive holds
    the carrier of R = union (I.:[#]R)
  proof
    let R be non empty RelStr,
        I be Function of the carrier of R, bool the carrier of R;
    assume AA: I is map-reflexive;
    thus the carrier of R c= union (I.:[#]R)
    proof
      let x be object;
      assume A0: x in the carrier of R; then
      reconsider y = x as Element of R;
A2:   y in I.x by AA;
      x in dom I by A0,FUNCT_2:def 1; then
      I.x in I.:[#]R by FUNCT_1:def 6;
      hence thesis by A2,TARSKI:def 4;
    end;
    let x be object;
    assume x in union (I.:[#]R); then
    consider y being set such that
T1: x in y & y in I.:[#]R by TARSKI:def 4;
    thus thesis by T1;
  end;
