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_an_SDR_of XX holds space(D) is_an_SDR_of
  space(TrivExt D)
proof
  let XX be non empty TopSpace, X be closed non empty SubSpace of XX, D be
  DECOMPOSITION of X;
  given CH1 being continuous Function of [:XX,I[01]:],XX such that
A1: for A being Point of XX holds CH1. [A,0[01]] =A & CH1. [A,1[01]] in
  the carrier of X & (A in the carrier of X implies for T being Point of I[01]
  holds CH1. [A,T] =A);
  the carrier of [:XX,I[01]:]=[:the carrier of XX, the carrier of I[01] :]
  by Def2;
  then reconsider
  II=[:Proj TrivExt D, id the carrier of I[01]:] as Function of the
  carrier of [:XX,I[01]:], the carrier of [:space TrivExt D,I[01]:] by Def2;
  now
    given xx,xx9 being Point of [:XX,I[01]:] such that
A2: II.xx=II.xx9 and
A3: (Proj TrivExt D*CH1).xx<>(Proj TrivExt D*CH1).xx9;
A4: (Proj TrivExt D*CH1).xx = Proj TrivExt D.(CH1.xx) & (Proj TrivExt D*
    CH1).xx9 = Proj TrivExt D.(CH1.xx9) by FUNCT_2:15;
    consider x being Point of XX, t being Point of I[01] such that
A5: xx=[x,t] by Th10;
    consider x9 being Point of XX, t9 being Point of I[01] such that
A6: xx9=[x9,t9] by Th10;
A7: II.(x,t)=[Proj TrivExt D.x,t] & II.(x9,t9)=[Proj TrivExt D.x9,t9] by
EQREL_1:58;
    then t=t9 & Proj TrivExt D.x=Proj TrivExt D.x9 by A2,A5,A6,XTUPLE_0:1;
    then CH1.xx=x & CH1.xx9=x9 by A1,A3,A5,A6,Th35;
    hence contradiction by A2,A3,A5,A6,A7,A4,XTUPLE_0:1;
  end;
  then consider
  THETA being Function of the carrier of [:space TrivExt D,I[01]:],
  the carrier of space TrivExt D such that
A8: Proj TrivExt D*CH1 = THETA*II by EQREL_1:56;
  reconsider THETA as Function of [:space TrivExt D,I[01]:], space TrivExt D;
  THETA is continuous
  proof
    let WT be Point of [:space TrivExt D,I[01]:];
    reconsider xt9=WT as Element of [:the carrier of space TrivExt D, the
    carrier of I[01]:] by Def2;
    let G be a_neighborhood of THETA.WT;
    reconsider FF = Proj TrivExt D*CH1 as continuous Function of [:XX,I[01]:],
    space TrivExt D;
    consider W being Point of space TrivExt D, T being Point of I[01] such
    that
A9: WT=[W,T] by Th10;
    II"{xt9} = [:Proj TrivExt D"{W},{T}:] by A9,EQREL_1:60;
    then reconsider
    GG=FF"G as a_neighborhood of [:Proj TrivExt D"{W},{T}:] by A8,Th4,
EQREL_1:57;
    W in the carrier of space TrivExt D;
    then
A10: W in TrivExt D by Def7;
    then (Proj TrivExt D)"{W} = W by EQREL_1:66;
    then Proj TrivExt D"{W} is compact by A10,Def12;
    then consider U being a_neighborhood of Proj TrivExt D"{W}, V being
    a_neighborhood of T such that
A11: [:U,V:] c= GG by Th25;
    reconsider H9=Proj(TrivExt D).:U as a_neighborhood of W by Th38;
    reconsider H99=[:H9,V:] as a_neighborhood of WT by A9;
    take H=H99;
    II.:[:U,V:]=[:Proj TrivExt D.:U,V:] by EQREL_1:59;
    then
A12: (Proj TrivExt D*CH1).:GG c= G & THETA.:H=(Proj TrivExt D*CH1).:[:U,V
    :] by A8,FUNCT_1:75,RELAT_1:126;
    (Proj TrivExt D*CH1).:[:U,V:] c= (Proj TrivExt D*CH1).:GG by A11,
RELAT_1:123;
    hence thesis by A12;
  end;
  then reconsider
  THETA9=THETA as continuous Function of [:space TrivExt D,I[01]:],
  space TrivExt D;
  take THETA99=THETA9;
  let W be Point of space(TrivExt D);
  consider W9 being Point of XX such that
A13: Proj(TrivExt D).W9=W by Th29;
  II.(W9,0[01]) = [W,0[01]] by A13,EQREL_1:58;
  then
A14: (THETA9*II). [W9,0[01]] = THETA9. [W,0[01]] by FUNCT_2:15;
  CH1.(W9,0[01]) =W9 by A1;
  hence THETA99. [W,0[01]] =W by A8,A13,A14,FUNCT_2:15;
A15: CH1. [W9,1[01]] in the carrier of X by A1;
  II.(W9,1[01]) =[W,1[01]] by A13,EQREL_1:58;
  then
A16: (THETA9*II). [W9,1[01]] = THETA9. [W,1[01]] by FUNCT_2:15;
  (THETA9*II). [W9,1[01]] = Proj TrivExt D.(CH1. [W9,1[01]]) by A8,FUNCT_2:15;
  hence THETA99. [W,1[01]] in the carrier of space(D) by A16,A15,Th37;
  assume
A17: W in the carrier of space(D);
  let T be Point of I[01];
  II.(W9,T) = [W,T] by A13,EQREL_1:58;
  then
A18: (THETA9*II). [W9,T] = THETA9. [W,T] by FUNCT_2:15;
  CH1.(W9,T) =W9 by A1,A13,A17,Th36;
  hence thesis by A8,A13,A18,FUNCT_2:15;
end;
