
theorem Th24:
for X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
 F be Finite_Sep_Sequence of S, a be FinSequence of ExtREAL,
 f be PartFunc of X,ExtREAL, g be PartFunc of Y,ExtREAL
  st T is bijective & g = f*T"
   & F,a are_Re-presentation_of f holds
   ex G be Finite_Sep_Sequence of CopyField(T,S)
    st G = ((.:T) |S) * F & G,a are_Re-presentation_of g
proof
    let X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
    F be Finite_Sep_Sequence of S, a be FinSequence of ExtREAL,
    f be PartFunc of X,ExtREAL, g be PartFunc of Y,ExtREAL;
    assume that
A1: T is bijective and
A2: g = f*T" and
A3: F,a are_Re-presentation_of f;

A4: dom T = X & rng T = Y by A1,FUNCT_2:def 1,def 3;
A5: rng T = dom(T") by A1,FUNCT_1:33;

    set H = (.:T) |S;
    rng H = (.:T).:S by RELAT_1:115; then
A6: rng H = CopyField(T,S) by A1,Def2;

A7: dom H = S by FUNCT_2:def 1; then
    reconsider H as Function of S,CopyField(T,S) by A6,FUNCT_2:1;
    reconsider G = H*F as Finite_Sep_Sequence of CopyField(T,S) by Th21,A1;
    take G;
    thus G = ((.:T) |S) * F;

A8: dom G = dom a by A3,A7,RELAT_1:27;

A9:for x be object holds x in dom g iff x in union (rng G)
    proof
     let x be object;
     hereby assume x in dom g; then
A10:   x in dom (T") & T".x in dom f by A2,FUNCT_1:11; then
      consider Y be set such that
A11:   T".x in Y & Y in (rng F) by A3,TARSKI:def 4;
      consider t be object such that
A12:  t in dom F & Y = F.t by A11,FUNCT_1:def 3;
      reconsider t as Nat by A12;
A13:  F.t in S;

A14:  x = T.(T".x) by A10,A1,A5,FUNCT_1:35;
      T.(T".x) in T.:Y by A10,A4,A11,FUNCT_1:def 6; then
      x in (.:T).Y by A1,A12,A13,A14,Th1; then
      x in H.Y by A12,A13,FUNCT_1:49; then
A15:  x in G.t by A12,FUNCT_1:13;

      t in dom G by A7,A12,A13,FUNCT_1:11; then
      G.t in rng G by FUNCT_1:3;
      hence x in union (rng G) by A15,TARSKI:def 4;
     end;
     assume x in union (rng G); then
     consider Y be set such that
A16: x in Y & Y in rng G by TARSKI:def 4;
     consider t be object such that
A17: t in dom G & Y = G.t by A16,FUNCT_1:def 3;

     reconsider t as Nat by A17;
     Y = H.(F.t) by A17,FUNCT_1:12; then
     Y = (.:T).(F.t) by FUNCT_1:49; then
     Y = T.:(F.t) by A1,Th1; then
     consider z be object such that
A18: z in dom T & z in (F.t) & x = T.z by A16,FUNCT_1:def 6;
A19: z = T".x by A18,A1,FUNCT_1:34;
A20: x in dom (T") by A5,A18,FUNCT_1:3;

     t in dom F & F.t in dom H by A17,FUNCT_1:11; then
     F.t in rng F by FUNCT_1:3; then
     z in union rng F by A18,TARSKI:def 4;
     hence x in dom g by A19,A3,A20,A2,FUNCT_1:11;
    end;

    for n being Nat st n in dom G holds
     for x being object st x in G.n holds g.x = a.n
    proof
     let n be Nat;
     assume
A21: n in dom G;
     let x be object;
     assume x in G.n; then
A22: x in H.(F.n) by FUNCT_1:12,A21;
     H.(F.n) = (.:T).(F.n) by FUNCT_1:49; then
     H.(F.n) = T.:(F.n) by A1,Th1; then
     consider t be object such that
A23: t in dom T & t in F.n & x = T.t by A22,FUNCT_1:def 6;
     reconsider t as Element of X by A23;

     g.x = f.(T".(T.t)) by A2,A4,A5,A23,FUNCT_1:13; then
     g.x = f.t by A23,A1,FUNCT_1:34;
     hence g.x = a.n by A23,A3,A8,A21;
    end;
    hence thesis by A9,A3,A7,RELAT_1:27,TARSKI:2;
end;
