reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem
  for XX being non empty TopSpace, X being closed non empty SubSpace of
  XX, D being DECOMPOSITION of X st X is_a_retract_of XX holds space D
  is_a_retract_of space TrivExt D
proof
  let XX be non empty TopSpace, X be closed non empty SubSpace of XX, D be
  DECOMPOSITION of X;
  thus X is_a_retract_of XX implies space(D) is_a_retract_of space(TrivExt D)
  proof
    given R being continuous Function of XX,X such that
A1: R is being_a_retraction;
    now
      given xx,xx9 being Point of XX such that
A2:   Proj TrivExt D.xx=Proj TrivExt D.xx9 & (Proj D*R).xx<>(Proj D*R) .xx9;
      xx in the carrier of X by A2,Th35;
      then R.xx=xx by A1;
      then
A3:   Proj D.xx = (Proj D*R).xx by FUNCT_2:15;
      xx9 in the carrier of X by A2,Th35;
      then
A4:   R.xx9=xx9 by A1;
      Proj TrivExt D.xx=Proj D.xx & Proj TrivExt D.xx9=Proj D.xx9 by A2,Th33
,Th35;
      hence contradiction by A2,A4,A3,FUNCT_2:15;
    end;
    then consider
    F being Function of the carrier of space TrivExt D, the carrier
    of space D such that
A5: F*(Proj TrivExt D)=(Proj D)*R by EQREL_1:56;
    reconsider F as Function of space TrivExt D, space D;
    F is continuous
    proof
      let W be Point of space TrivExt D;
      let G be a_neighborhood of F.W;
      reconsider GG=(Proj D*R)"G as a_neighborhood of Proj TrivExt D"{W} by A5
,Th4,EQREL_1:57;
      reconsider V9=Proj TrivExt D.:GG as a_neighborhood of W by Th38;
      take V=V9;
      F.:V=(Proj D*R).:GG by A5,RELAT_1:126;
      hence thesis by FUNCT_1:75;
    end;
    then reconsider F9=F as continuous Function of space TrivExt D, space D;
    take F9;
    let W be Point of space TrivExt D;
    consider W9 being Point of XX such that
A6: Proj TrivExt D.W9=W by Th29;
    assume
A7: W in the carrier of space D;
    then W9 in the carrier of X by A6,Th36;
    then
A8: W9=R.W9 by A1;
A9: F9.W = (F*Proj TrivExt D).W9 by A6,FUNCT_2:15;
    Proj D.W9=Proj TrivExt D.W9 by A6,A7,Th33,Th36;
    hence thesis by A5,A6,A8,A9,FUNCT_2:15;
  end;
end;
