
theorem Th28:
  for I be non empty set,
      J be non-empty disjoint_valued ManySortedSet of I,
      F be Group-Family of I,J,
      x be Function
  holds support(x,Union F) = Union supp_restr(x,F)
  proof
    let I be non empty set,
        J be non-empty disjoint_valued ManySortedSet of I,
        F be Group-Family of I,J,
        x be Function;
    set y = supp_restr(x,F);
    A1: dom J = I by PARTFUN1:def 2;
    A2: dom y = I by PARTFUN1:def 2;
    for j be object holds j in support(x,Union F) iff j in Union y
    proof
      let j be object;
      hereby
        assume j in support(x,Union F); then
        consider Z being Group such that
        A3: Z = (Union F).j & x.j <> 1_Z & j in Union J by GROUP_19:def 1;
        consider Y be set such that
        A4: j in Y & Y in rng J by A3,TARSKI:def 4;
        consider i being object such that
        A5: i in dom J & Y = J.i by A4,FUNCT_1:def 3;
        reconsider i as Element of I by A5;
        reconsider j0 = j as Element of J.i by A4,A5;
        (F.i).j0 = (Union F).j0 by Th19; then
        (x | (J.i)).j <> 1_(F.i).j0 by A3,FUNCT_1:49; then
        j in support (x | (J.i),F.i) by GROUP_19:def 1; then
        A6: j in y.i by Def12;
        y.i in rng y by A2,FUNCT_1:3;
        hence j in Union y by A6,TARSKI:def 4;
      end;
      assume j in Union y; then
      consider Y be set such that
      A7: j in Y & Y in rng y by TARSKI:def 4;
      consider i being object such that
      A8: i in dom y & Y = y.i by A7,FUNCT_1:def 3;
      reconsider i as Element of I by A8;
      A9: y.i = support(x | (J.i), F.i) by Def12; then
      consider Z being Group such that
      A10: Z = (F.i).j & (x | (J.i)).j <> 1_Z & j in J.i
      by A7,A8,GROUP_19:def 1;
      A11: (x | (J.i)).j = x.j by A7,A8,A9,FUNCT_1:49;
      J.i in rng J by A1,FUNCT_1:3;
      then reconsider j0 = j as Element of Union J by A10,TARSKI:def 4;
      reconsider ZZ = (Union F).j0 as Group;
      1_Z = 1_ZZ by A10,Th19;
      hence j in support(x,Union F) by A10,A11,GROUP_19:def 1;
    end;
    hence thesis by TARSKI:2;
  end;
