reserve X, Y for non empty set;

theorem
  for r being Relation of X st r is_reflexive_in X holds chi(r,[:X,X:])
  is reflexive
proof
  let r be Relation of X;
  assume
A1: r is_reflexive_in X;
  for x being Element of X holds chi(r,[:X,X:]).(x,x) = 1
  proof
    let x be Element of X;
    [x,x] in r by A1,RELAT_2:def 1;
    hence thesis by FUNCT_3:def 3;
  end;
  hence thesis;
end;
