reserve X,Y for set,
  x,x1,x2,y,y1,y2,z for set,
  f,g,h for Function;
reserve M for non empty set;
reserve D for non empty set;
reserve P for Relation;
reserve O for Order of X;
reserve R,P for Relation,
  X,X1,X2,Y,Z,x,y,z,u for set,
  g,h for Function,
  O for Order of X,
  D for non empty set,
  d,d1,d2 for Element of D,
  A1,A2,B for Ordinal,
  L,L1,L2 for Sequence;

theorem Th14:
  dom O = X & rng O = X
proof
  thus dom O = X
  proof
    thus dom O c= X;
    let x be object;
    assume x in X;
    then [x,x] in O by Th3;
    hence thesis by XTUPLE_0:def 12;
  end;
  thus rng O c= X;
  let x be object;
  assume x in X;
  then [x,x] in O by Th3;
  hence thesis by XTUPLE_0:def 13;
end;
