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 Th13:
  for X being non empty functional compatible set holds
  rng union X = union the set of all rng f where f is Element of X
proof
  let X be non empty functional compatible set;
  set F = the set of all rng f where f is Element of X;
  now
    let y be object;
    hereby
      assume y in rng union X;
      then consider x being object such that
A1:   [x,y] in union X by XTUPLE_0:def 13;
      consider Z being set such that
A2:   [x,y] in Z and
A3:   Z in X by A1,TARSKI:def 4;
      reconsider Z as Element of X by A3;
A4:   rng Z in F;
      y in rng Z by A2,XTUPLE_0:def 13;
      hence y in union F by A4,TARSKI:def 4;
    end;
    assume y in union F;
    then consider Z being set such that
A5: y in Z and
A6: Z in F by TARSKI:def 4;
    consider f being Element of X such that
A7: Z = rng f by A6;
    consider x being object such that
A8: [x, y] in f by A5,A7,XTUPLE_0:def 13;
    [x, y] in union X by A8,TARSKI:def 4;
    hence y in rng union X by XTUPLE_0:def 13;
  end;
  hence thesis by TARSKI:2;
end;
