
theorem
  for T being non empty TopSpace, T0 being T_0-TopSpace, f being
continuous Function of T,T0 ex h being continuous Function of T_0-reflex(T),T0
  st f = h*T_0-canonical_map(T)
proof
  let T be non empty TopSpace;
  let T0 be T_0-TopSpace;
  let f be continuous Function of T,T0;
  set F = T_0-canonical_map(T);
  set R = Indiscernibility(T);
  set TR = T_0-reflex(T);
  defpred X[object,object] means ex D1 being set st D1 = $1 & $2 in f.:D1;
A1: for C being object st C in the carrier of TR
    ex y being object st y in the carrier of T0 & X[C,y]
  proof
    let C be object;
    assume C in the carrier of TR;
    then consider p being Point of T such that
A2: C = Class(R,p) by Th3;
A3: f.p in {f.p} by TARSKI:def 1;
    reconsider C as set by TARSKI:1;
    f.:C = {f.p} by A2,Th12;
    hence thesis by A3;
  end;
  ex h being Function of the carrier of TR,the carrier of T0 st for C
  being object st C in the carrier of TR holds X[C,h.C]
from FUNCT_2:sch 1(A1);
  then consider h being Function of the carrier of TR,the carrier of T0
   such that
A4: for C being object st C in the carrier of TR holds X[C,h.C];
A5: for p being Point of T holds h.Class(R,p) = f.p
  proof
    let p be Point of T;
    Class(R,p) is Point of TR by Th3;
    then X[Class(R,p),h.Class(R,p)] by A4;
    then h.Class(R,p) in f.:Class(R,p);
    then h.Class(R,p) in {f.p} by Th12;
    hence thesis by TARSKI:def 1;
  end;
  reconsider h as Function of TR,T0;
A6: [#]T0 <> {};
  for W being Subset of T0 st W is open holds h"W is open
  proof
    let W be Subset of T0;
    assume W is open;
    then
A7: f"W is open by A6,TOPS_2:43;
    set V = h"W;
    for x being object holds x in union V iff x in f"W
    proof
      let x be object;
      hereby
        assume x in union V;
        then consider C being set such that
A8:     x in C and
A9:     C in V by TARSKI:def 4;
        consider p being Point of T such that
A10:    C = Class(R,p) by A9,Th3;
        x in the carrier of T by A8,A10;
        then
A11:    x in dom f by FUNCT_2:def 1;
        [x,p] in R by A8,A10,EQREL_1:19;
        then
A12:    C = Class(R,x) by A8,A10,EQREL_1:35;
        h.C in W by A9,FUNCT_1:def 7;
        then f.x in W by A5,A8,A12;
        hence x in f"W by A11,FUNCT_1:def 7;
      end;
      assume
A13:  x in f"W;
      then f.x in W by FUNCT_1:def 7;
      then
A14:  h.Class(R,x) in W by A5,A13;
      Class(R,x) is Point of TR by A13,Th3;
      then
A15:  Class(R,x) in V by A14,FUNCT_2:38;
      x in Class(R,x) by A13,EQREL_1:20;
      hence thesis by A15,TARSKI:def 4;
    end;
    then union V = f"W by TARSKI:2;
    then union V in the topology of T by A7;
    hence thesis by Th2;
  end;
  then reconsider h as continuous Function of TR,T0 by A6,TOPS_2:43;
  set H = h*F;
  for x being object st x in the carrier of T holds f.x = H.x
  proof
    let x be object;
    assume
A16: x in the carrier of T;
    then Class(R,x) in Class R by EQREL_1:def 3;
    then
A17: Class(R,x) in the carrier of TR by BORSUK_1:def 7;
    x in dom F & F.x = Class(R,x) by A16,Th4,FUNCT_2:def 1;
    then
A18:  (h*F).x = h.Class(R,x) by FUNCT_1:13;
    X[Class(R,x),h.Class(R,x)] by A4,A17;
    then H.x in f.:Class(R,x) by A18;
    then H.x in {f.x} by A16,Th12;
    hence thesis by TARSKI:def 1;
  end;
  hence thesis by FUNCT_2:12;
end;
