reserve X for non empty TopSpace;
reserve X for non empty TopSpace;
reserve X for non empty TopSpace,
  X0 for non empty maximal_Kolmogorov_subspace of X;

theorem Th22:
  ex r being continuous Function of X,X0 st r is being_a_retraction
proof
  reconsider M = the carrier of X0 as Subset of X by TSEP_1:1;
  defpred X[Point of X,set] means M /\ MaxADSet $1 = {$2};
A1: M is maximal_T_0 by Th11;
A2: for x being Point of X ex a being Point of X0 st X[x,a]
  proof
    let x be Point of X;
    consider a being Point of X such that
A3: a in M and
A4: M /\ MaxADSet(x) = {a} by A1;
    reconsider a as Point of X0 by A3;
    take a;
    thus thesis by A4;
  end;
  consider r being Function of X,X0 such that
A5: for x being Point of X holds X[x,r.x] from FUNCT_2:sch 3(A2);
  reconsider r as continuous Function of X,X0 by A5,Th18;
  take r;
  thus thesis by A5,Th20;
end;
