
theorem
  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 such that
A1: for i being Element of I holds A.i is strongly_connected;
  let B be Segre-Coset of A;
  let f be Collineation of Segre_Product A;
  reconsider g=f" as Collineation of Segre_Product A by Th13;
A2: f is bijective by Def4;
  then rng f = [#](Segre_Product A) by FUNCT_2:def 3;
  then f"B = g.:B by A2,TOPS_2:55;
  hence thesis by A1,Th24;
end;
