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 Th10:
  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 M.Omega < +infty & len x = card (
Omega) & s is one-to-one & rng s = Omega & len s = card (Omega) & (for n being
Nat st n in dom x
   holds x.n =  (f.(s.n) qua ExtReal) * M.{s.n})
  holds Integral(M,f) = Sum x
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: M.Omega < +infty and
A2: len x = card (Omega) and
A3: s is one-to-one and
A4: rng s = Omega and
A5: len s = card (Omega) and
A6: for n being Nat st n in dom x
   holds x.n = (f.(s.n) qua ExtReal) * M.{s.n};
  set Sigma= Trivial-SigmaField (Omega);
  consider F be Finite_Sep_Sequence of Sigma, a being FinSequence of REAL such
  that
A7: dom f = union (rng F) and
  dom a = dom s and
A8: dom F = dom s and
A9: for k be Nat st k in dom F holds F.k={s.k} and
  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 by A3,A4,Th9;
A10: dom x =Seg len s by A2,A5,FINSEQ_1:def 3
    .= dom F by A8,FINSEQ_1:def 3;
  set fm=max-(f);
  set fp=max+(f);
A11: dom f =Omega by FUNCT_2:def 1;
  then dom fp = Omega by RFUNCT_3:def 10;
  then
A12: fp is_integrable_on M by A1,Lm5,Th6;
A13: for n be Nat st n in dom s holds M.{s.n} in REAL
  proof
    let n be Nat;
    assume n in dom s;
    then s.n in rng s by FUNCT_1:3;
    then {s.n } c= Omega by ZFMISC_1:31;
    hence M.({s.n }) in REAL by A1,A4,Lm7;
  end;
  now
    let y be object;
    assume y in rng x;
    then consider n be Element of NAT such that
A14: n in dom x and
A15: y=x.n by PARTFUN1:3;
    reconsider z=M.{s.n} as Element of REAL by A8,A10,A13,A14;
    reconsider w=f.(s.n) as Element of REAL by XREAL_0:def 1;
    x.n = (f.(s.n) ) * M.{s.n} by A6,A14
      .= w*z by EXTREAL1:1;
    hence y in REAL by A15;
  end;
  then rng x c= REAL;
  then reconsider x1=x as FinSequence of REAL by FINSEQ_1:def 4;
A16: fm is_simple_func_in Sigma & fm is nonnegative by Th6,MESFUNC6:61;
  defpred AP[Nat,set] means for x be object st x in F.$1 holds $2=fp.x;
  set L=len F;
A17: dom F =Seg L by FINSEQ_1:def 3;
A18: for k be Nat st k in Seg L ex y being Element of REAL st AP[k,y]
  proof
    let k be Nat;
    assume
A19: k in Seg L;
    reconsider fpsk = fp.(s.k) as Element of REAL by XREAL_0:def 1;
    take fpsk;
    F.k = {s.k} by A9,A17,A19;
    hence thesis by TARSKI:def 1;
  end;
  consider ap being FinSequence of REAL such that
A20: dom ap = Seg L & for k be Nat st k in Seg L holds AP[k,ap.k] from
  FINSEQ_1:sch 5(A18);
  defpred AM[Nat,set] means for x be object st x in F.$1 holds $2=fm.x;
A21: for k be Nat st k in Seg L ex y being Element of REAL st AM[k,y]
  proof
    let k be Nat;
    assume
A22: k in Seg L;
      reconsider fmsk = fm.(s.k) as Element of REAL by XREAL_0:def 1;
    take fmsk;
    F.k = {s.k} by A9,A17,A22;
    hence thesis by TARSKI:def 1;
  end;
  consider am being FinSequence of REAL such that
A23: dom am = Seg L & for k be Nat st k in Seg L holds AM[k,am.k] from
  FINSEQ_1:sch 5(A21);
A24: dom fm=dom f by RFUNCT_3:def 11;
  then
A25: fm is Function of Omega,REAL by A11,FUNCT_2:def 1;
  then M.(dom ((-1)(#)fm)) < +infty by A1,FUNCT_2:def 1;
  then (-1)(#)fm is_integrable_on M by A25,Lm5,Th6;
  then consider E be Element of Sigma such that
A26: E = (dom fp) /\ dom ((-1)(#)fm) and
A27: Integral(M,fp+(-1)(#)fm) =Integral(M,fp|E)+Integral(M,((-1)(#)fm)|E
  ) by A12,MESFUNC6:101;
A28: (-jj)*Integral(M,fm)
     =(- (jj qua ExtReal)) *Integral(M,fm)
    .= -((1 qua ExtReal)*Integral(M,fm) ) by XXREAL_3:92
    .=-Integral(M,fm) by XXREAL_3:81;
  defpred Pp[ Nat,set ] means $2= (ap.$1) *(M*F).$1;
A29: for k being Nat st k in Seg L holds ex x being Element of ExtREAL st Pp
  [k,x];
  consider xp being FinSequence of ExtREAL such that
A30: dom xp = Seg L & for k being Nat st k in Seg L holds Pp[k,xp.k]
  from FINSEQ_1:sch 5(A29);
A31: dom xp = dom F by A30,FINSEQ_1:def 3;
A32: for n being Nat st n in dom xp holds xp.n = (fp.(s.n) ) * M.{s.n}
  proof
    let n be Nat;
    assume
A33: n in dom xp;
    then
A34: (M*F).n =M.(F.n) by A31,FUNCT_1:13
      .= M.{s.n} by A9,A31,A33;
    F.n={s.n} by A9,A31,A33;
    then
A35: s.n in F.n by TARSKI:def 1;
    thus xp.n= (ap.n) *(M*F).n by A30,A33
      .=  (fp.(s.n) ) * M.{s.n} by A20,A30,A33,A35,A34;
  end;
  now
    let y be object;
    assume y in rng xp;
    then consider n be Element of NAT such that
A36: n in dom xp and
A37: y=xp.n by PARTFUN1:3;
    reconsider z=M.{s.n} as Element of REAL by A8,A31,A13,A36;
    reconsider w=fp.(s.n) as Element of REAL by XREAL_0:def 1;
    xp.n =  (fp.(s.n) ) * M.{s.n} by A32,A36
      .= w*z by EXTREAL1:1;
    hence y in REAL by A37;
  end;
  then rng xp c= REAL;
  then reconsider xp1=xp as FinSequence of REAL by FINSEQ_1:def 4;
  defpred Pm[ Nat,set ] means $2= (am.$1) *(M*F).$1;
A38: for k being Nat st k in Seg L holds ex x being Element of ExtREAL st Pm
  [k,x];
  consider xm being FinSequence of ExtREAL such that
A39: dom xm = Seg L & for k being Nat st k in Seg L holds Pm[k,xm.k]
  from FINSEQ_1:sch 5(A38);
A40: dom xm = dom F by A39,FINSEQ_1:def 3;
A41: for n being Nat st n in dom xm holds xm.n = (fm.(s.n) ) * M.{s.n}
  proof
    let n be Nat;
    assume
A42: n in dom xm;
    then
A43: (M*F).n =M.(F.n) by A40,FUNCT_1:13
      .= M.{s.n} by A9,A40,A42;
    F.n={s.n} by A9,A40,A42;
    then
A44: s.n in F.n by TARSKI:def 1;
    thus xm.n= (am.n) *(M*F).n by A39,A42
      .= (fm.(s.n) ) * M.{s.n} by A23,A39,A42,A44,A43;
  end;
  now
    let y be object;
    assume y in rng xm;
    then consider n be Element of NAT such that
A45: n in dom xm and
A46: y=xm.n by PARTFUN1:3;
    reconsider z=M.{s.n} as Element of REAL by A8,A40,A13,A45;
    reconsider w=fm.(s.n) as Element of REAL by XREAL_0:def 1;
    xm.n = (fm.(s.n) ) * M.{s.n} by A41,A45
      .= w*z by EXTREAL1:1;
    hence y in REAL by A46;
  end;
  then rng xm c= REAL;
  then reconsider xm1=xm as FinSequence of REAL by FINSEQ_1:def 4;
A47: Sum(xp) = Sum(xp1) & Sum(xm) = Sum(xm1) by MESFUNC3:2;
A48: for k be Nat st k in dom x1 holds (xp1 - xm1).k =x1.k
  proof
    let k be Nat;
A49: f =fp-fm by MESFUNC6:42;
    assume
A50: k in dom x1;
    then reconsider z=M.{s.k} as Element of REAL by A8,A10,A13;
A51: xm1.k= (fm.(s.k) ) * M.{s.k} by A10,A40,A41,A50
      .= (fm.(s.k) ) * z by EXTREAL1:1;
    s.k in rng s by A8,A10,A50,FUNCT_1:3;
    then s.k in Omega;
    then
A52: s.k in dom f by FUNCT_2:def 1;
    k in (dom xp1) /\ (dom xm1) by A10,A31,A40,A50;
    then
A53: k in dom (xp1 - xm1) by VALUED_1:12;
    xp1.k= (fp.(s.k) ) * M.{s.k} by A10,A31,A32,A50
      .=(fp.(s.k) )*z by EXTREAL1:1;
    hence (xp1 - xm1).k = (fp.(s.k) ) * z - (fm.(s.k) ) * z by A53,A51,
VALUED_1:13
      .=( fp.(s.k) - (fm.(s.k)qua Real) ) * z
      .= (f.(s.k) ) * z by A52,A49,VALUED_1:13
      .= (f.(s.k) ) * M.{s.k}by EXTREAL1:1
      .=x1.k by A6,A50;
  end;
  dom fm = Omega by A11,RFUNCT_3:def 11;
  then
A54: fm is_integrable_on M by A1,Lm5,Th6;
A55: dom fp=dom f by RFUNCT_3:def 10;
A56: dom (xp1-xm1) = (dom xp1) /\ (dom xm1) by VALUED_1:12
    .= dom x1 by A10,A31,A40;
A57: len xp1=L by A30,FINSEQ_1:def 3
    .= len xm1 by A39,FINSEQ_1:def 3;
  dom ((-1)(#)fm) = dom fm by VALUED_1:def 5
    .=Omega by A11,RFUNCT_3:def 11;
  then
A58: E = Omega /\ Omega by A11,A26,RFUNCT_3:def 10
    .= Omega;
A59: Integral(M,fp+(-1)(#)fm) =Integral(M,fp) +Integral(M,(-1)(#)fm) by A27
,A58
    .=Integral(M,fp) +-Integral(M,fm) by A54,A28,MESFUNC6:102;
  fp is_simple_func_in Sigma & fp is nonnegative by Th6,MESFUNC6:61;
  then
A60: Integral(M,fp)=Sum(xp) by A7,A55,A17,A20,A30,Lm1;
  thus Integral(M,f) =Integral(M,fp-fm) by MESFUNC6:42
    .=Sum(xp) -Sum(xm) by A7,A24,A17,A23,A16,A39,A60,A59,Lm1
    .=Sum(xp1) -Sum(xm1) by A47,SUPINF_2:3
    .=Sum(xp1-xm1) by A57,INTEGRA5:2
    .=Sum(x) by A56,A48,FINSEQ_1:13,MESFUNC3:2;
end;
