
theorem Ble1:
  for L being non empty 1-sorted,
      R being total Relation of the carrier of L holds
    R is reflexive iff for x being Element of L holds [x,x] in R
  proof
    let L be non empty 1-sorted,
        R be total Relation of the carrier of L;
    thus R is reflexive implies for x being Element of L holds [x,x] in R
    proof
      assume
A1:   R is reflexive;
      let x be Element of L;
A2:   R is_reflexive_in field R by A1,RELAT_2:def 9;
      dom R = the carrier of L by PARTFUN1:def 2; then
      x in dom R \/ rng R by XBOOLE_0:def 3;
      hence thesis by A2,RELAT_2:def 1;
    end;
    assume
B1: for x being Element of L holds [x,x] in R;
    for x being object st x in field R holds [x,x] in R
    proof
      let x be object;
b2:   field R c= (the carrier of L) \/ the carrier of L by RELSET_1:8;
      assume x in field R;
      hence thesis by B1,b2;
    end;
    hence thesis by RELAT_2:def 9,RELAT_2:def 1;
  end;
