
theorem Th24:
  for I being non empty set for A being PLS-yielding ManySortedSet
  of I for x being set holds x is Block of Segre_Product A iff ex L being
Segre-like non trivial-yielding ManySortedSubset of Carrier A st x = product L
  & L.indx(L) is Block of A.indx(L)
proof
  let I be non empty set;
  let A be PLS-yielding ManySortedSet of I;
  let x be set;
  thus x is Block of Segre_Product A implies ex L being Segre-like non
trivial-yielding ManySortedSubset of Carrier A st x = product L & L.indx(L) is
  Block of A.indx(L)
  proof
    assume
A1: x is Block of Segre_Product A;
    then consider L being Segre-like ManySortedSubset of Carrier A such that
A2: x = product L and
A3: ex i being Element of I st L.i is Block of A.i by Def22;
    2 c= card (product L) by A1,A2,Def6;
    then reconsider L as Segre-like non trivial-yielding ManySortedSubset of
    Carrier A by Th13;
    consider i being Element of I such that
A4: L.i is Block of A.i by A3;
    now
      assume i <> indx(L);
      then
A5:   L.i is 1-element by Th12;
      2 c= card (L.i) by A4,Def6;
      hence contradiction by A5,Th4;
    end;
    hence thesis by A2,A4;
  end;
  given L being Segre-like non trivial-yielding ManySortedSubset of Carrier A
  such that
A6: x = product L & L.indx(L) is Block of A.indx(L);
  thus thesis by A6,Def22;
end;
