
theorem Th17:
  for I being non empty set for A being non-Empty
TopStruct-yielding ManySortedSet of I st for i being Element of I holds A.i is
  with_non_trivial_blocks & ex i being Element of I st A.i is non void holds
  Segre_Product A is with_non_trivial_blocks
proof
  let I be non empty set;
  let A be non-Empty TopStruct-yielding ManySortedSet of I;
  assume
A1: for i being Element of I holds A.i is with_non_trivial_blocks & ex i
  being Element of I st A.i is non void;
  for k being Block of Segre_Product A holds 2 c= card k
  proof
    let k be Block of Segre_Product A;
    Segre_Product A is non void by A1,Th15;
    then consider B being Segre-like ManySortedSubset of Carrier A such that
A2: k = product B and
A3: ex i being Element of I st B.i is Block of A.i by Def22;
    consider i being Element of I such that
A4: B.i is Block of A.i by A3;
    A.i is with_non_trivial_blocks by A1;
    then 2 c= card (B.i) by A4;
    then
A5: B.i is non trivial by Th4;
    dom B = I by PARTFUN1:def 2;
    then B.i in rng B by FUNCT_1:def 3;
    then reconsider
    BB=B as Segre-like non trivial-yielding ManySortedSet of I by A5,Def16;
    product BB is non trivial;
    hence thesis by A2,Th4;
  end;
  hence thesis;
end;
