
theorem Th23:
  for I be non empty set,
      J be non-empty disjoint_valued ManySortedSet of I,
      F be Group-Family of I,J,
      y be Element of product(Union F),
      i be Element of I
  holds y | (J.i) in product(F.i)
  proof
    let I be non empty set,
        J be non-empty disjoint_valued ManySortedSet of I,
        F be Group-Family of I,J,
        y be Element of product(Union F),
        i be Element of I;
    set x = y | (J.i);
    A1: dom J = I by PARTFUN1:def 2;
    A2: dom(Union F) = Union J by PARTFUN1:def 2;
    A3: dom y = Union J by GROUP_19:3;
    A4: J.i c= Union J by A1,FUNCT_1:3,ZFMISC_1:74; then
    A5: dom x = J.i by A3,RELAT_1:62;
    set z = Carrier(F.i);
    A6: dom z = J.i by PARTFUN1:def 2;
    for j be object st j in J.i holds x.j in z.j
    proof
      let j be object;
      assume j in J.i; then
      reconsider j as Element of J.i;
      reconsider j1 = j as Element of Union J by A4;
      reconsider D = Union F as Group-Family of Union J;
      y in product(Union F); then
      A8: y.j1 in D.j1 by GROUP_19:5;
      A9: x.j = y.j by FUNCT_1:49;
      [j, (Union F).j] in Union F by A2,A4,FUNCT_1:1; then
      consider Y0 be set such that
      A10: [j, (Union F).j] in Y0 & Y0 in rng F by TARSKI:def 4;
      consider k being object such that
      A11: k in dom F & Y0 = F.k by A10,FUNCT_1:def 3;
      reconsider k as Element of I by A11;
      reconsider Fk = F.k as Function;
      A12: dom Fk = J.k by PARTFUN1:def 2;
      j in dom Fk by A10,A11,XTUPLE_0:def 12; then
      A13: J.k /\ J.i <> {} by A12,XBOOLE_0:def 4;
      A14: i = k by A13,PROB_2:def 2,XBOOLE_0:def 7;
      reconsider T = (F.i).j as Group;
      z.j = [#]T by PENCIL_3:7;
      hence thesis by A8,A9,A10,A11,A14,FUNCT_1:1;
    end; then
    x in product z by A5,A6,CARD_3:def 5;
    hence thesis by GROUP_7:def 2;
  end;
