
theorem Th16:
  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
  non degenerated & ex i being Element of I st A.i is non void holds
  Segre_Product A is non degenerated
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 non degenerated & ex i being
  Element of I st A.i is non void;
  then Segre_Product A is non void by Th15;
  then reconsider SB=Segre_Blocks A as non empty set;
  now
    assume product Carrier A in SB;
    then consider B being Segre-like ManySortedSubset of Carrier A such that
A2: product Carrier A = 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;
    B is non-empty by A2,Th1;
    then
    (ex R being 1-sorted st R=A.i & the carrier of R = ( Carrier A).i )& (
    Carrier A).i is Block of A.i by A2,A4,PRALG_1:def 15,PUA2MSS1:2;
    then A.i is degenerated;
    hence contradiction by A1;
  end;
  then not product Carrier A is Element of SB;
  hence thesis;
end;
