reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;
reserve x,y,z for set,
  S, X for non empty set,
  R for Relation of X;

theorem
  R is irreflexive & field R c= S implies R is_irreflexive_in S
proof
  assume that
A1: R is irreflexive and
A2: field R c= S;
  let x be object;
  S = field R \/ ( S \ (field R) ) by A2,XBOOLE_1:45;
  then
A3: x in S implies x in field R or x in S \ (field R) by XBOOLE_0:def 3;
A4: x in S \ (field R) implies not [x,x] in R
  proof
    assume x in S \ (field R);
    then x in S \ (dom R \/ rng R) by RELAT_1:def 6;
    then x in (S \ dom R) /\ (S \ rng R) by XBOOLE_1:53;
    then x in (S \ rng R) by XBOOLE_0:def 4;
    then not x in rng R by XBOOLE_0:def 5;
    hence thesis by XTUPLE_0:def 13;
  end;
  R is_irreflexive_in field R by A1;
  hence thesis by A3,A4;
end;
