
theorem
  for S, T being non empty TopSpace st the TopStruct of S = the
  TopStruct of T & S is sober holds T is sober
proof
  let S, T be non empty TopSpace such that
A1: the TopStruct of S = the TopStruct of T and
A2: for A be irreducible Subset of S ex a being Point of S st a
is_dense_point_of A & for b being Point of S st b is_dense_point_of A holds a =
  b;
  let B be irreducible Subset of T;
  reconsider A = B as irreducible Subset of S by A1,Th23;
  consider a being Point of S such that
A3: a is_dense_point_of A and
A4: for b being Point of S st b is_dense_point_of A holds a = b by A2;
  reconsider p = a as Point of T by A1;
  take p;
  thus p is_dense_point_of B by A1,A3,Th24;
  let q be Point of T;
  reconsider b = q as Point of S by A1;
  assume q is_dense_point_of B;
  then b is_dense_point_of A by A1,Th24;
  hence thesis by A4;
end;
