
theorem Th8:
  for I being non empty set for A being non-Trivial-yielding
TopStruct-yielding ManySortedSet of I for B1,B2 being Segre-Coset of A st 2 c=
  card(B1 /\ B2) holds B1 = B2
proof
  let I be non empty set;
  let A be non-Trivial-yielding TopStruct-yielding ManySortedSet of I;
  let B1,B2 be Segre-Coset of A;
  consider L1 being Segre-like non trivial-yielding ManySortedSubset of
  Carrier A such that
A1: B1 = product L1 and
A2: L1.indx(L1) = [#](A.indx(L1)) by Def2;
  consider L2 being Segre-like non trivial-yielding ManySortedSubset of
  Carrier A such that
A3: B2 = product L2 and
A4: L2.indx(L2) = [#](A.indx(L2)) by Def2;
  assume
A5: 2 c= card(B1 /\ B2);
  then
A6: indx L1 = indx L2 by A1,A3,PENCIL_1:26;
A7: now
    let i be object;
    assume i in I;
    per cases;
    suppose
      i <> indx L1;
      hence L1.i = L2.i by A5,A1,A3,PENCIL_1:26;
    end;
    suppose
      i = indx(L1);
      hence L1.i = L2.i by A2,A4,A6;
    end;
  end;
A8: dom L2 = I by PARTFUN1:def 2;
  dom L1 = I by PARTFUN1:def 2;
  hence thesis by A1,A3,A8,A7,FUNCT_1:2;
end;
