
theorem Th24:
  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
  Segre-Coset of A for f being Collineation of Segre_Product A holds f.:B is
  Segre-Coset of A
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 Segre-Coset of A;
  consider W being Subset of Segre_Product A such that
A2: W is non trivial strong closed_under_lines and
A3: B = union {Y where Y is Subset of Segre_Product A : Y is non trivial
  strong closed_under_lines & W,Y are_joinable} by A1,Th23;
  let f be Collineation of Segre_Product A;
  reconsider g=f" as Collineation of Segre_Product A by Th13;
A4: dom f = the carrier of Segre_Product A by FUNCT_2:def 1;
A5: dom g = the carrier of Segre_Product A by FUNCT_2:def 1;
A6: f is bijective by Def4;
  then
A7: rng f = the carrier of Segre_Product A by FUNCT_2:def 3;
A8: rng f = [#](Segre_Product A) by A6,FUNCT_2:def 3;
A9: f.:B = union {Y where Y is Subset of Segre_Product A : Y is non trivial
  strong closed_under_lines & f.:W,Y are_joinable}
  proof
A10: W c= f"(f.:W) by A4,FUNCT_1:76;
    f"(f.:W) c= W by A6,FUNCT_1:82;
    then
A11: f"(f.:W) = W by A10;
    thus f.:B c= union {Y where Y is Subset of Segre_Product A: Y is non
    trivial strong closed_under_lines & f.:W,Y are_joinable}
    proof
      let o be object;
      assume o in f.:B;
      then consider u being object such that
A12:  u in dom f and
A13:  u in B and
A14:  o = f.u by FUNCT_1:def 6;
      consider y being set such that
A15:  u in y and
A16:  y in {Y where Y is Subset of Segre_Product A : Y is non trivial
      strong closed_under_lines & W,Y are_joinable} by A3,A13,TARSKI:def 4;
      consider Y being Subset of Segre_Product A such that
A17:  y=Y and
A18:  Y is non trivial strong closed_under_lines and
A19:  W,Y are_joinable by A16;
A20:  f.:W,f.:Y are_joinable by A2,A18,A19,Th20;
      f.:Y is non trivial strong closed_under_lines by A18,Th14,Th16,Th18;
      then
A21:  f.:Y in {Z where Z is Subset of Segre_Product A: Z is non trivial
      strong closed_under_lines & f.:W,Z are_joinable} by A20;
      o in f.:Y by A12,A14,A15,A17,FUNCT_1:def 6;
      hence thesis by A21,TARSKI:def 4;
    end;
    let o be object;
    assume o in union {Y where Y is Subset of Segre_Product A : Y is non
    trivial strong closed_under_lines & f.:W,Y are_joinable};
    then consider u being set such that
A22: o in u and
A23: u in {Y where Y is Subset of Segre_Product A : Y is non trivial
    strong closed_under_lines & f.:W,Y are_joinable} by TARSKI:def 4;
    consider Y being Subset of Segre_Product A such that
A24: u=Y and
A25: Y is non trivial strong closed_under_lines and
A26: f.: W,Y are_joinable by A23;
A27: g.:Y is non trivial strong closed_under_lines by A25,Th14,Th16,Th18;
    f.:W is non trivial strong closed_under_lines by A2,Th14,Th16,Th18;
    then g.:(f.:W),g.:Y are_joinable by A25,A26,Th20;
    then W,g.:Y are_joinable by A6,A8,A11,TOPS_2:55;
    then
A28: g.:Y in {Z where Z is Subset of Segre_Product A : Z is non trivial
    strong closed_under_lines & W,Z are_joinable} by A27;
    g.o in g.:Y by A5,A22,A24,FUNCT_1:def 6;
    then
A29: g.o in B by A3,A28,TARSKI:def 4;
    o = f.((f qua Function)".o) by A6,A7,A22,A24,FUNCT_1:35;
    then o = f.(g.o) by A6,TOPS_2:def 4;
    hence thesis by A4,A29,FUNCT_1:def 6;
  end;
  f.:W is non trivial strong closed_under_lines by A2,Th14,Th16,Th18;
  hence thesis by A1,A9,Th23;
end;
