
theorem Th10:
  for I being non empty set for A being non-Trivial-yielding
  TopStruct-yielding ManySortedSet of I for b being Segre-like non
trivial-yielding ManySortedSubset of Carrier A st product b is Segre-Coset of A
  holds b.indx(b) = [#](A.indx(b))
proof
  let I be non empty set;
  let A be non-Trivial-yielding TopStruct-yielding ManySortedSet of I;
  let b be Segre-like non trivial-yielding ManySortedSubset of Carrier A;
  assume product b is Segre-Coset of A;
  then consider
  L being Segre-like non trivial-yielding ManySortedSubset of Carrier
  A such that
A1: product b = product L and
A2: L.indx(L) = [#](A.indx(L)) by PENCIL_2:def 2;
  b=L by A1,PUA2MSS1:2;
  hence thesis by A2;
end;
