
theorem
  for X being set, R being Relation, Y being Subset of X st
    R is_irreflexive_in X holds R is_irreflexive_in Y
proof
  let X be set, R be Relation, Y be Subset of X;
  assume A1: R is_irreflexive_in X;
  for x being object holds x in Y implies not [x,x] in R by A1, RELAT_2:def 2;
  hence thesis by RELAT_2:def 2;
end;
