
theorem LEM3b:
  for X being non empty set,
      R being total reflexive Relation of X holds
    R /\ R~ is total
  proof
    let X be non empty set,
        R be total reflexive Relation of X;
A4: field R = X by FieldTot; then
A5: field (R~) = X by RELAT_1:21;
A6: id X c= R by A4,RELAT_2:1;
    id field (R~) c= R~ by RELAT_2:1; then
    id X c= R /\ R~ by XBOOLE_1:19, A6,A5; then
    dom (id X) c= dom (R /\ R~) by RELAT_1:11;
    hence thesis by PARTFUN1:def 2,XBOOLE_0:def 10;
  end;
