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_in X implies R is irreflexive
proof
A1: field R c= X \/ X by RELSET_1:8;
  assume R is_irreflexive_in X;
  then for x being object holds x in field R implies not [x,x] in R by A1;
  then R is_irreflexive_in field R;
  hence thesis;
end;
