reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th14:
  for P being compatible PFUNC_DOMAIN of X,Y holds union P is PartFunc of X,Y
proof
  let D be compatible PFUNC_DOMAIN of X,Y;
  set E = the set of all dom f where f is Element of D;
  set F = the set of all rng f where f is Element of D;
  reconsider u = union D as Function;
A1: rng u c= Y
  proof
    let y be object;
    assume y in rng u;
    then y in union F by Th13;
    then consider Z being set such that
A2: y in Z and
A3: Z in F by TARSKI:def 4;
    consider f being Element of D such that
A4: Z = rng f by A3;
    rng f c= Y by RELAT_1:def 19;
    hence thesis by A2,A4;
  end;
  dom u c= X
  proof
    let x be object;
    assume x in dom u;
    then x in union E by Th11;
    then consider Z being set such that
A5: x in Z and
A6: Z in E by TARSKI:def 4;
    ex f being Element of D st Z = dom f by A6;
    hence thesis by A5;
  end;
  hence thesis by A1,RELSET_1:4;
end;
