
theorem Th23:
  for I being non empty set for A being PLS-yielding ManySortedSet
  of I st for i being Element of I holds A.i is strongly_connected for B being
  set holds B is Segre-Coset of A iff ex W being Subset of Segre_Product A st W
  is non trivial strong closed_under_lines & B = union {Y where Y is Subset of
Segre_Product A : Y is non trivial strong closed_under_lines & W,Y are_joinable
  }
proof
  let I be non empty set;
  let A be PLS-yielding ManySortedSet of I;
  assume
A1: for i being Element of I holds A.i is strongly_connected;
  let B be set;
  thus B is Segre-Coset of A implies ex W being Subset of Segre_Product A st W
  is non trivial strong closed_under_lines & B = union {Y where Y is Subset of
Segre_Product A : Y is non trivial strong closed_under_lines & W,Y are_joinable
  }
  proof
    assume B is Segre-Coset of A;
    then reconsider BB=B as Segre-Coset of A;
    consider L being Segre-like non trivial-yielding ManySortedSubset of
    Carrier A such that
A2: BB = product L and
A3: L.indx(L) = [#](A.indx(L)) by Def2;
    set K1 = the Block of A.indx(L);
    consider V being object such that
A4: V in product L by XBOOLE_0:def 1;
    K1 in the topology of A.indx(L);
    then reconsider K = K1 as Subset of A.indx(L);
A5: ex g being Function st g = V & dom g = dom L &
for i being object st i
    in dom L holds g.i in L.i by A4,CARD_3:def 5;
A6: dom L = I by PARTFUN1:def 2;
    then reconsider V as ManySortedSet of I by A5,PARTFUN1:def 2,RELAT_1:def 18
;
    for i being object st i in I holds V.i is Element of (Carrier A).i
    proof
      let i be object;
      assume
A7:   i in I;
      L c= Carrier A by PBOOLE:def 18;
      then
A8:   L.i c= (Carrier A).i by A7;
      V.i in L.i by A6,A5,A7;
      hence thesis by A8;
    end;
    then reconsider V as Element of Carrier A by PBOOLE:def 14;
    reconsider VV={V} as ManySortedSubset of Carrier A by PENCIL_1:11;
    reconsider X=VV+*(indx(L),K) as ManySortedSubset of Carrier A by Th7;
    2 c= card K by PENCIL_1:def 6;
    then
A9: K is non trivial by PENCIL_1:4;
    then reconsider X as Segre-like non trivial-yielding ManySortedSubset of
    Carrier A by PENCIL_1:9,10;
    dom VV = I by PARTFUN1:def 2;
    then
A10: X.indx(L) = K by FUNCT_7:31;
    then indx(X) = indx(L) by A9,PENCIL_1:def 21;
    then reconsider pX = product X as Block of Segre_Product A by A10,
PENCIL_1:24;
A11: for i being object st i in I holds X.i c= L.i
    proof
      let i be object;
      assume
A12:  i in I;
      per cases;
      suppose
        i <> indx L;
        then X.i = VV.i by FUNCT_7:32;
        then
A13:    X.i = {V.i} by A12,PZFMISC1:def 1;
        V.i in L.i by A6,A5,A12;
        hence thesis by A13,ZFMISC_1:31;
      end;
      suppose
        i = indx(L);
        hence thesis by A3,A10;
      end;
    end;
    pX in the topology of Segre_Product A;
    then reconsider psX = pX as Subset of Segre_Product A;
    take psX;
    thus
A14: psX is non trivial strong closed_under_lines by PENCIL_1:20,21;
    then reconsider
    Z = union {Y where Y is Subset of Segre_Product A : Y is non
trivial strong closed_under_lines & psX,Y are_joinable} as Segre-Coset of A by
A1,Th22;
    psX,psX are_joinable by A14,Th10;
    then
A15: psX in {Y where Y is Subset of Segre_Product A : Y is non trivial
    strong closed_under_lines & psX,Y are_joinable} by A14;
A16: psX c= Z
    by A15,TARSKI:def 4;
    dom X = I by PARTFUN1:def 2;
    then psX c= B by A2,A6,A11,CARD_3:27;
    then psX c= (B /\ Z) by A16,XBOOLE_1:19;
    then
A17: card psX c= card (B /\ Z) by CARD_1:11;
    2 c= card pX by PENCIL_1:def 6;
    then BB=Z by A17,Th8,XBOOLE_1:1;
    hence thesis;
  end;
  given W being Subset of Segre_Product A such that
A18: W is non trivial strong closed_under_lines and
A19: B = union {Y where Y is Subset of Segre_Product A : Y is non
  trivial strong closed_under_lines & W,Y are_joinable};
  thus thesis by A1,A18,A19,Th22;
end;
