
theorem Th19:
  for I be non empty set,
      J be disjoint_valued ManySortedSet of I,
      F be Group-Family of I,J,
      j be Element of I,
      i be object st i in J.j
  holds (Union F).i = (F.j).i
  proof
    let I be non empty set,
        J be disjoint_valued ManySortedSet of I,
        F be Group-Family of I,J,
        j be Element of I,
        i be object;
    assume
    A1: i in J.j;
    dom J = I by PARTFUN1:def 2; then
    A3: J.j c= Union J by FUNCT_1:3,ZFMISC_1:74;
    dom(Union F) = Union J by PARTFUN1:def 2; then
    [i, (Union F).i] in Union F by A1,A3,FUNCT_1:1; then
    consider Y0 be set such that
    A4: [i, (Union F).i] in Y0 & Y0 in rng F by TARSKI:def 4;
    consider k being object such that
    A5: k in dom F & Y0 = F.k by A4,FUNCT_1:def 3;
    reconsider k as Element of I by A5;
    reconsider Fk = F.k as Function;
    A6: dom Fk = J.k by PARTFUN1:def 2;
    i in dom Fk by A4,A5,XTUPLE_0:def 12; then
    J.k /\ J.j <> {} by A1,A6,XBOOLE_0:def 4; then
    j = k by PROB_2:def 2,XBOOLE_0:def 7;
    hence thesis by A4,A5,FUNCT_1:1;
  end;
