
theorem Th21:
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 st T is bijective holds
  ((.:T) |S)*F is Finite_Sep_Sequence of CopyField(T,S)
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;
    assume
A1: T is bijective;

    set H = (.:T) |S;
    rng H = (.:T).:S by RELAT_1:115; then
A2: rng H = CopyField(T,S) by A1,Def2;

    dom H = S by FUNCT_2:def 1; then
    reconsider H as Function of S,CopyField(T,S) by A2,FUNCT_2:1;
    reconsider G = H*F as FinSequence of CopyField(T,S);

    for m, n being object st m <> n holds G.m misses G.n
    proof
     let m,n be object;
     assume
A3:  m <> n;

     per cases;
     suppose not m in dom G or not n in dom G; then
      G.m = {} or G.n = {} by FUNCT_1:def 2;
      hence G.m misses G.n;
     end;
     suppose
A4:   m in dom G & n in dom G; then
      reconsider m1 = m, n1 = n as Element of NAT;

      G.n1 = H.(F.n1) by A4,FUNCT_1:12; then
      G.n1 =(.:T).(F.n1) by FUNCT_1:49; then
A5:   G.n1 =T.:(F.n1) by A1,Th1;

      G.m1 = H.(F.m1) by A4,FUNCT_1:12; then
      G.m1 = (.:T).(F.m1) by FUNCT_1:49; then
A6:   G.m1 =T.:(F.m1) by A1,Th1;

      F is disjoint_valued; then
      F.m misses F.n by A3; then
      {} = T.: (F.m1 /\ F.n1);
      hence G.m misses G.n by A1,A5,A6,FUNCT_1:62;
     end;
    end; then
    G is disjoint_valued;
    hence thesis;
end;
