reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;

theorem Th9:
  for Omega be non empty finite set, M being sigma_Measure of
Trivial-SigmaField (Omega), f be Function of Omega,REAL, x being FinSequence of
ExtREAL, s being FinSequence of (Omega) st s is one-to-one & rng s = Omega ex F
  be Finite_Sep_Sequence of Trivial-SigmaField (Omega), a being FinSequence of
  REAL st dom f = union (rng F) & dom a = dom s & dom F = dom s & (for k be Nat
  st k in dom F holds F.k={s.k} ) & for n being Nat for x,y being Element of
  Omega st n in dom F & x in F.n & y in F.n holds f.x = f.y
proof
  let Omega be non empty finite set, M be sigma_Measure of Trivial-SigmaField
  (Omega), f be Function of Omega,REAL, x be FinSequence of ExtREAL, s be
  FinSequence of (Omega);
  assume that
A1: s is one-to-one and
A2: rng s = Omega;
  defpred Q[Nat,set] means $2=f.(s.$1);
  set Sigma=Trivial-SigmaField (Omega);
  set L = len s;
  defpred P[Nat,set] means $2={s.$1};
A3: for k be Nat st k in Seg L ex x being Element of bool Omega st P[k,x]
  proof
    let k be Nat;
    assume
A4: k in Seg L;
    take {s.k};
    k in dom s by A4,FINSEQ_1:def 3;
    then s.k in rng s by FUNCT_1:3;
    hence thesis by ZFMISC_1:31;
  end;
  consider F being FinSequence of bool Omega such that
A5: dom F = Seg L & for k be Nat st k in Seg L holds P[k,F.k] from
  FINSEQ_1:sch 5(A3);
A6: now
    let i,j be Nat;
    assume that
A7: i in dom F & j in dom F and
A8: i <> j;
    i in dom s & j in dom s by A5,A7,FINSEQ_1:def 3;
    then
A9: s.i <> s.j by A1,A8,FUNCT_1:def 4;
    F.i = {s.i } & F.j = {s.j } by A5,A7;
    hence F.i misses F.j by A9,ZFMISC_1:11;
  end;
A10: dom F = dom s by A5,FINSEQ_1:def 3;
  reconsider F as Finite_Sep_Sequence of Sigma by A6,MESFUNC3:4;
  union rng F = rng s
  proof
    now
      let x be object;
      assume x in union rng F;
      then consider y be set such that
A11:  x in y and
A12:  y in rng F by TARSKI:def 4;
      consider n be object such that
A13:  n in dom F and
A14:  y=F.n by A12,FUNCT_1:def 3;
      F.n = {s.n } by A5,A13;
      then
A15:  x = s.n by A11,A14,TARSKI:def 1;
      n in dom s by A5,A13,FINSEQ_1:def 3;
      hence x in rng s by A15,FUNCT_1:def 3;
    end;
    hence union rng F c= rng s;
    let x be object;
    assume x in rng s;
    then consider n be object such that
A16: n in dom s and
A17: x=s.n by FUNCT_1:def 3;
A18: n in Seg L by A16,FINSEQ_1:def 3;
    reconsider n as Element of NAT by A16;
    n in dom F by A5,A16,FINSEQ_1:def 3;
    then
A19: F.n in rng F by FUNCT_1:def 3;
    x in {s.n} by A17,TARSKI:def 1;
    then x in F.n by A5,A18;
    hence x in union rng F by A19,TARSKI:def 4;
  end;
  then
A20: dom f = union rng F by A2,FUNCT_2:def 1;
  take F;
A21: for n being Nat for x,y being Element of Omega st n in dom F & x in F.n
  & y in F.n holds f.x = f.y
  proof
    let n be Nat;
    let x,y be Element of Omega;
    assume that
A22: n in dom F and
A23: x in F.n and
A24: y in F.n;
A25: F.n = {s.n} by A5,A22;
    hence f.x =f.(s.n) by A23,TARSKI:def 1
      .=f.y by A24,A25,TARSKI:def 1;
  end;
A26: for k be Nat st k in Seg L ex x being Element of REAL st Q[k,x]
   proof let k be Nat;
     f.(s.k) in REAL by XREAL_0:def 1;
    hence thesis;
   end;
  ex a being FinSequence of REAL st ( dom a = Seg L & for k be Nat st k in
  Seg L holds Q[k,a.k]) from FINSEQ_1:sch 5(A26 );
  hence thesis by A5,A10,A20,A21;
end;
