
theorem Th15:
  for I being non empty set for A being non-Empty
TopStruct-yielding ManySortedSet of I st ex i being Element of I st A.i is non
  void holds Segre_Product A is non void
proof
  let I be non empty set;
  let A be non-Empty TopStruct-yielding ManySortedSet of I;
  set B = the trivial-yielding non-empty ManySortedSubset of Carrier A;
  given i being Element of I such that
A1: A.i is non void;
  set l = the Block of A.i;
A2: the topology of A.i is non empty by A1;
A3: B+*(i,l) c= Carrier A
  proof
    let i1 be object;
    assume
A4: i1 in I;
    then
A5: i1 in dom B by PARTFUN1:def 2;
    per cases;
    suppose
A6:   i = i1;
      then B+*(i,l).i1 = l by A5,FUNCT_7:31;
      then
A7:   B+*(i,l).i1 in the topology of (A.i) by A2;
      ex R being 1-sorted st R=A.i1 & the carrier of R = ( Carrier A).i1
      by A4,PRALG_1:def 15;
      hence thesis by A6,A7;
    end;
    suppose
A8:   i1 <> i;
A9:   B c= Carrier A by PBOOLE:def 18;
      B+*(i,l).i1 = B.i1 by A8,FUNCT_7:32;
      hence thesis by A4,A9;
    end;
  end;
  for j being Element of I st i<>j holds B+*(i,l).j is 1-element
  proof
    let j be Element of I;
    assume i<>j;
    then
A10: B+*(i,l).j = B.j by FUNCT_7:32;
    j in I;
    then
A11: j in dom B by PARTFUN1:def 2;
    then B+*(i,l).j in rng B by A10,FUNCT_1:def 3;
    then B+*(i,l).j is non empty trivial by A11,A10,Def16,FUNCT_1:def 9;
    hence thesis;
  end;
  then reconsider C=B+*(i,l) as Segre-like ManySortedSubset of Carrier A by A3
,Def20,PBOOLE:def 18;
  dom B = I by PARTFUN1:def 2;
  then C.i is Block of A.i by FUNCT_7:31;
  then product C in Segre_Blocks A by Def22;
  hence thesis;
end;
