reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem
  E = dom(F.0) & F.0 is nonnegative & F is with_the_same_dom & (for n be
  Nat holds F.n is E-measurable) & (for n,m be Nat st n <=m holds for x be
Element of X st x in E holds (F.n).x <= (F.m).x ) & (for x be Element of X st x
in E holds F#x is convergent) implies ex I be ExtREAL_sequence st (for n be Nat
  holds I.n = Integral(M,F.n)) & I is convergent & Integral(M,lim F) = lim I
proof
  assume that
A1: E = dom(F.0) and
A2: F.0 is nonnegative and
A3: F is with_the_same_dom and
A4: for n be Nat holds F.n is E-measurable and
A5: for n,m be Nat st n <= m holds for x be Element of X st x in E holds
  (F.n).x <= (F.m).x and
A6: for x be Element of X st x in E holds F#x is convergent;
A7: lim F is E-measurable by A1,A3,A4,A6,MESFUNC8:25;
A8: for n be Nat holds F.n is nonnegative
  proof
    let n be Nat;
    for x be object st x in dom(F.n) holds 0 <= (F.n).x
    proof
      let x be object;
      assume x in dom(F.n);
      then x in E by A1,A3;
      then (F.0).x <= (F.n).x by A5;
      hence thesis by A2,SUPINF_2:51;
    end;
    hence thesis by SUPINF_2:52;
  end;
  per cases;
  suppose
    ex n be Nat st M.(E /\ eq_dom(F.n,+infty)) <> 0;
    then consider N be Nat such that
A9: M.(E /\ eq_dom(F.N,+infty)) <> 0;
A10: E = dom(F.N) by A1,A3;
    then reconsider EE = E /\ eq_dom(F.N,+infty) as Element of S by A4,
MESFUNC1:33;
A11: EE c= E by XBOOLE_1:17;
    then
A12: F.N is EE-measurable by A4,MESFUNC1:30;
    EE c= dom(F.N) by A1,A3,A11;
    then
A13: EE = dom((F.N)|EE) by RELAT_1:62;
    then EE = dom(F.N) /\ EE by RELAT_1:61;
    then
A14: (F.N)|EE is EE-measurable by A12,MESFUNC5:42;
    now
      let x be object;
      assume
A15:  x in EE;
      then x in eq_dom(F.N,+infty) by XBOOLE_0:def 4;
      then (F.N).x = +infty by MESFUNC1:def 15;
      then ((F.N)|EE).x = +infty by A13,A15,FUNCT_1:47;
      hence x in eq_dom((F.N)|EE,+infty) by A13,A15,MESFUNC1:def 15;
    end;
    then
A16: EE c= eq_dom((F.N)|EE,+infty);
    for x be object st x in eq_dom((F.N)|EE,+infty) holds x in EE by A13,
MESFUNC1:def 15;
    then eq_dom((F.N)|EE,+infty) c= EE;
    then EE = eq_dom((F.N)|EE,+infty) by A16,XBOOLE_0:def 10;
    then
A17: M.(EE /\ eq_dom((F.N)|EE,+infty)) <> 0 by A9;
    F.N is E-measurable by A4;
    then
A18: Integral(M,(F.N)|EE) <= Integral(M,(F.N)|E) by A8,A10,A11,MESFUNC5:93;
    reconsider N1 = N as Element of NAT by ORDINAL1:def 12;
    deffunc I1(Element of NAT) = Integral(M,F.$1);
    consider I be sequence of ExtREAL such that
A19: for n be Element of NAT holds I.n = I1(n) from FUNCT_2:sch 4;
    reconsider I as ExtREAL_sequence;
A20: 0 < M.(E /\ eq_dom(F.N,+infty)) by A9,SUPINF_2:51;
A21: for n be Nat holds I.n = Integral(M,F.n)
    proof
      let n be Nat;
      reconsider n1=n as Element of NAT by ORDINAL1:def 12;
      I.n = Integral(M,F.n1) by A19;
      hence thesis;
    end;
    take I;
A22: dom(lim F) = dom(F.0) by MESFUNC8:def 9;
    for x be object st x in dom(lim F) holds (lim F).x >= 0
    proof
      let x be object;
      assume
A23:  x in dom(lim F);
      then reconsider x1=x as Element of X;
      for n be Nat holds (F#x1).n >= 0
      proof
        let n be Nat;
A24:    (F.0).x1 >= 0 by A2,SUPINF_2:51;
        (F.n).x1 >= (F.0).x1 by A1,A5,A22,A23;
        hence thesis by A24,MESFUNC5:def 13;
      end;
      then lim(F#x1) >= 0 by A1,A6,A22,A23,Th10;
      hence thesis by A23,MESFUNC8:def 9;
    end;
    then
A25: lim F is nonnegative by SUPINF_2:52;
A26: E = dom(lim F) by A1,MESFUNC8:def 9;
A27: EE c= E /\ eq_dom(lim F,+infty)
    proof
      let x be object;
      assume
A28:  x in EE;
      then reconsider x1=x as Element of X;
      x in eq_dom(F.N,+infty) by A28,XBOOLE_0:def 4;
      then (F.N).x1 = +infty by MESFUNC1:def 15;
      then
A29:  (F#x1).N = +infty by MESFUNC5:def 13;
A30:  x in E by A28,XBOOLE_0:def 4;
      for n,m be Nat st m <= n holds (F#x1).m <= (F#x1).n
      proof
        let n,m be Nat;
        assume m <= n;
        then (F.m).x1 <= (F.n).x1 by A5,A30;
        then (F#x1).m <= (F.n).x1 by MESFUNC5:def 13;
        hence thesis by MESFUNC5:def 13;
      end;
      then
A31:  (F#x1) is non-decreasing by RINFSUP2:7;
      then
A32:  (F#x1)^\N1 is non-decreasing by RINFSUP2:26;
      ((F#x1)^\N1).0 = (F#x1).(0+N) by NAT_1:def 3;
      then for n be Element of NAT holds +infty <= ((F#x1)^\N1).n by A29,A32,
RINFSUP2:7;
      then (F#x1)^\N1 is convergent_to_+infty by RINFSUP2:32;
      then
A33:  lim((F#x1)^\N1) = +infty by Th7;
A34:  sup(F#x1) = sup((F#x1)^\N1) by A31,RINFSUP2:26;
      lim(F#x1) = sup(F#x1) by A31,RINFSUP2:37;
      then lim(F#x1) = +infty by A32,A34,A33,RINFSUP2:37;
      then (lim F).x1 = +infty by A1,A22,A30,MESFUNC8:def 9;
      then x in eq_dom(lim F,+infty) by A26,A30,MESFUNC1:def 15;
      hence thesis by A30,XBOOLE_0:def 4;
    end;
A35: for n,m be Nat st m<=n holds I.m<=I.n
    proof
      let n,m be Nat;
A36:  F.m is E-measurable by A4;
      assume m <= n;
      then
A37:  for x be Element of X st x in E holds (F.m).x <= (F.n).x by A5;
A38:  E = dom(F.m) by A1,A3;
A39:  E = dom(F.n) by A1,A3;
A40:    n in NAT by ORDINAL1:def 12;
A41:    m in NAT by ORDINAL1:def 12;
      F.n is E-measurable by A4;
      then Integral(M,(F.m)|E) <= Integral(M,(F.n)|E) by A8,A38,A39,A36,A37
,Th15;
      then Integral(M,F.m) <= Integral(M,(F.n)|E) by A38;
      then Integral(M,F.m) <= Integral(M,F.n) by A39;
      then I.m <= Integral(M,F.n) by A19,A41;
      hence thesis by A19,A40;
    end;
    then
A42: I is non-decreasing by RINFSUP2:7;
    then
A43: I^\N1 is non-decreasing by RINFSUP2:26;
    F.N is nonnegative by A8;
    then Integral(M,(F.N)|EE) = +infty by A13,A14,A17,Th13,MESFUNC5:15;
    then +infty <= Integral(M,F.N) by A10,A18;
    then
A44: Integral(M,F.N) = +infty by XXREAL_0:4;
    for k be Element of NAT holds +infty <= (I^\N1).k
    proof
      let k be Element of NAT;
      I.N1 <= I.(N1+k) by A35,NAT_1:12;
      then I.N1 <= (I^\N1).k by NAT_1:def 3;
      hence thesis by A44,A21;
    end;
    then I^\N1 is convergent_to_+infty by RINFSUP2:32;
    then
A45: lim(I^\N1) = +infty by Th7;
    E /\ eq_dom(lim F,+infty) is Element of S by A7,A26,MESFUNC1:33;
    then
A46: M.(E /\ eq_dom(lim F,+infty)) <> 0 by A27,A20,MEASURE1:31;
A47: sup I = sup(I^\N1) by A42,RINFSUP2:26;
    lim I = sup I by A42,RINFSUP2:37;
    then lim I = +infty by A43,A47,A45,RINFSUP2:37;
    hence thesis by A7,A21,A42,A26,A25,A46,Th13,RINFSUP2:37;
  end;
  suppose
A48: for n be Nat holds M.(E /\ eq_dom(F.n,+infty)) = 0;
    defpred L[Element of NAT,set] means $2 = E /\ eq_dom(F.$1,+infty);
A49: for n be Element of NAT ex A be Element of S st L[n,A]
    proof
      let n be Element of NAT;
      E c= dom(F.n) by A1,A3;
      then reconsider A = E /\ eq_dom(F.n,+infty) as Element of S by A4,
MESFUNC1:33;
      take A;
      thus thesis;
    end;
    consider L be sequence of S such that
A50: for n be Element of NAT holds L[n,L.n] from FUNCT_2:sch 3(A49);
A51: rng L c= S by RELAT_1:def 19;
    rng L is N_Sub_set_fam of X by MEASURE1:23;
    then reconsider E0 = rng L as N_Measure_fam of S by A51,MEASURE2:def 1;
    set E1 = E\(union E0);
    deffunc H(Nat) = (F.$1)|E1;
    consider H be Functional_Sequence of X,ExtREAL such that
A52: for n be Nat holds H.n = H(n) from SEQFUNC:sch 1;
    deffunc I2(Element of NAT) = Integral(M,(F.$1)|E1);
    consider I be sequence of ExtREAL such that
A53: for n be Element of NAT holds I.n = I2(n) from FUNCT_2:sch 4;
    reconsider I as ExtREAL_sequence;
A54: E1 c= E by XBOOLE_1:36;
    then
A55: for n be Nat holds F.n is E1-measurable by A4,MESFUNC1:30;
A56: for n be Nat holds dom(H.n) = E1 & H.n is without-infty & H.n is
    without+infty
    proof
      let n be Nat;
A57:  dom(H.n) = dom((F.n)|E1) by A52;
      E1 c= dom(F.n) by A1,A3,A54;
      hence dom(H.n) = E1 by A57,RELAT_1:62;
      (F.n)|E1 is nonnegative by A8,MESFUNC5:15;
      then H.n is nonnegative by A52;
      hence H.n is without-infty by MESFUNC5:12;
      for x be set st x in dom(H.n) holds (H.n).x < +infty
      proof
        reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
        let x be set;
A58:    L.n = E /\ eq_dom(F.n1,+infty) by A50;
        dom L = NAT by FUNCT_2:def 1;
        then
A59:    L.n in rng L by A58,FUNCT_1:3;
        assume x in dom(H.n);
        then
A60:    x in dom((F.n)|E1) by A52;
        then
A61:    x in E1 by RELAT_1:57;
A62:    x in dom(F.n) by A60,RELAT_1:57;
        assume
A63:    (H.n).x >= +infty;
        (H.n).x = ((F.n)|E1).x by A52;
        then (H.n).x = (F.n).x by A61,FUNCT_1:49;
        then (F.n).x = +infty by A63,XXREAL_0:4;
        then x in eq_dom(F.n,+infty) by A62,MESFUNC1:def 15;
        then x in L.n by A54,A61,A58,XBOOLE_0:def 4;
        then x in union E0 by A59,TARSKI:def 4;
        hence contradiction by A61,XBOOLE_0:def 5;
      end;
      hence thesis by MESFUNC5:11;
    end;
    for n,m be Nat holds dom(H.n) = dom(H.m)
    proof
      let n,m be Nat;
      dom(H.n) = E1 by A56;
      hence thesis by A56;
    end;
    then reconsider
    H as with_the_same_dom Functional_Sequence of X,ExtREAL by MESFUNC8:def 2;
    defpred G[Nat,set,set] means $3 = H.($1+1) - H.($1);
A64: for n being Nat for x being set ex y being set st G[n,x,y ];
    consider G being Function such that
A65: dom G = NAT & G.0 = H.0 & for n being Nat holds G[n,G
    .n,G.(n+1)] from RECDEF_1:sch 1(A64);
A66: for n be Nat holds G.(n+1) = H.(n+1)-H.n
    by A65;
    now
      defpred IND[Nat] means G.$1 is PartFunc of X,ExtREAL;
      let f be object;
      assume f in rng G;
      then consider m be object such that
A67:  m in dom G and
A68:  f = G.m by FUNCT_1:def 3;
      reconsider m as Nat by A65,A67;
A69:  for n be Nat st IND[n] holds IND[n+1]
      proof
        let n be Nat;
        assume IND[n];
        G.(n+1) = H.(n+1) - H.n by A66;
        hence thesis;
      end;
A70:  IND[ 0 ] by A65;
      for n be Nat holds IND[n] from NAT_1:sch 2(A70,A69);
      then G.m is PartFunc of X,ExtREAL;
      hence f in PFuncs(X,ExtREAL) by A68,PARTFUN1:45;
    end;
    then rng G c= PFuncs(X,ExtREAL);
    then reconsider G as Functional_Sequence of X,ExtREAL by A65,FUNCT_2:def 1
,RELSET_1:4;
A71: for n be Nat holds dom(G.n) = E1
    proof
      let n be Nat;
      now
        assume n <> 0;
        then consider k be Nat such that
A72:    n = k+1 by NAT_1:6;
A73:    H.(k+1) is without-infty by A56;
A74:    H.k is without+infty by A56;
        G.(k+1) = H.(k+1) - H.k by A66;
        then dom(G.(k+1)) = dom(H.(k+1)) /\ dom(H.k) by A73,A74,MESFUNC5:17;
        then dom(G.(k+1)) = E1 /\ dom(H.k) by A56;
        then dom(G.(k+1)) = E1 /\ E1 by A56;
        hence thesis by A72;
      end;
      hence thesis by A56,A65;
    end;
A75: for n,m be Nat holds dom(G.n) = dom(G.m)
    proof
      let n,m be Nat;
      dom(G.n) = E1 by A71;
      hence thesis by A71;
    end;
A76: for n be Nat holds G.n is nonnegative
    proof
      let n be Nat;
A77:  n <> 0 implies G.n is nonnegative
      proof
        assume n <> 0;
        then consider k be Nat such that
A78:    n = k + 1 by NAT_1:6;
A79:    G.(k+1) = H.(k+1) - H.k by A66;
        for x be object st x in dom(G.(k+1)) holds 0 <= (G.(k+1)).x
        proof
          let x be object;
          assume
A80:      x in dom(G.(k+1));
A81:      dom(G.(k+1)) = E1 by A71;
          (H.k).x = ((F.k)|E1).x by A52;
          then
A82:      (H.k).x = (F.k).x by A80,A81,FUNCT_1:49;
          (H.(k+1)).x = ((F.(k+1))|E1).x by A52;
          then
A83:      (H.(k+1)).x = (F.(k+1)).x by A80,A81,FUNCT_1:49;
          (F.k).x <= (F.(k+1)).x by A5,A54,A80,A81,NAT_1:11;
          then (H.(k+1)).x - (H.k).x >= 0 by A83,A82,XXREAL_3:40;
          hence thesis by A79,A80,MESFUNC1:def 4;
        end;
        hence thesis by A78,SUPINF_2:52;
      end;
      n = 0 implies G.n is nonnegative
      proof
        assume
A84:    n = 0;
        (F.n)|E1 is nonnegative by A8,MESFUNC5:15;
        hence thesis by A52,A65,A84;
      end;
      hence thesis by A77;
    end;
A85: for n1 be object st n1 in NAT holds H.n1 = (Partial_Sums G).n1
    proof
      defpred PH[Nat] means H.$1 = (Partial_Sums G).$1;
      let n1 be object;
      assume n1 in NAT;
      then reconsider n = n1 as Nat;
A86:  for k be Nat st PH[k] holds PH[k+1]
      proof
        let k be Nat;
A87:    H.k is without+infty by A56;
A88:    H.k is without-infty by A56;
A89:    dom(G.(k+1)) = E1 by A71;
        G.(k+1) is without-infty by A76,MESFUNC5:12;
        then dom(G.(k+1) + H.k) = dom(G.(k+1)) /\ dom(H.k) by A88,MESFUNC5:16;
        then dom(G.(k+1) + H.k) = E1 /\ E1 by A56,A89;
        then
A90:    dom(H.(k+1)) = dom(G.(k+1) + H.k) by A56;
A91:    G.(k+1) = H.(k+1) - H.k by A66;
        for x being Element of X st x in dom(H.(k+1)) holds (H.(k+1)).x
        = (G.(k+1) + H.k).x
        proof
          let x be Element of X;
A92:      (H.k).x <> +infty by A87;
          (H.k).x <> -infty by A88;
          then ( (H.(k+1)).x - (H.k).x ) + (H.k).x = (H.(k+1)).x - ( (H.k).x
          - (H.k).x ) by A92,XXREAL_3:32;
          then
          ( (H.(k+1)).x - (H.k).x ) + (H.k).x = (H.(k+1)).x - 0. by XXREAL_3:7;
          then
A93:      ( (H.(k+1)).x - (H.k).x ) + (H.k).x = (H.(k+1)).x by XXREAL_3:4;
          assume
A94:      x in dom(H.(k+1));
          then x in E1 by A56;
          then (H.(k+1)).x = (G.(k+1)).x + (H.k).x by A91,A89,A93,
MESFUNC1:def 4;
          hence thesis by A90,A94,MESFUNC1:def 3;
        end;
        then
A95:    H.(k+1) = G.(k+1) + H.k by A90,PARTFUN1:5;
        assume PH[k];
        hence thesis by A95,Def4;
      end;
A96:  PH[ 0 ] by A65,Def4;
      for k be Nat holds PH[k] from NAT_1:sch 2(A96,A86);
      then H.n = (Partial_Sums G).n;
      hence thesis;
    end;
    then
A97: for n be Nat holds H.n = (Partial_Sums G).n & lim H = lim(
    Partial_Sums G) by FUNCT_2:12;
    reconsider G as with_the_same_dom Functional_Sequence of X,ExtREAL by A75,
MESFUNC8:def 2;
    reconsider G as additive with_the_same_dom Functional_Sequence of X,
    ExtREAL by A76,Th30;
A98: for k be Nat holds H.k is real-valued
    proof
      let k be Nat;
      for x be Element of X st x in dom(H.k) holds |. (H.k).x .| < +infty
      proof
        let x be Element of X;
        assume x in dom(H.k);
        H.k is without+infty by A56;
        then
A99:    (H.k).x < +infty;
        H.k is without-infty by A56;
        then -infty < (H.k).x;
        hence thesis by A99,EXTREAL1:40,XXREAL_0:4;
      end;
      hence thesis by MESFUNC2:def 1;
    end;
A100: for n be Nat holds G.n is E1-measurable
    proof
      let n be Nat;
      n <> 0 implies G.n is E1-measurable
      proof
        assume n <> 0;
        then consider k be Nat such that
A101:    n = k + 1 by NAT_1:6;
A102:   E1 = dom(H.k) by A56;
A103:   G.(k+1) = H.(k+1) - H.k by A66;
A104:   H.k is real-valued by A98;
A105:   H.k is E1-measurable by A1,A3,A55,A52,Th20,XBOOLE_1:36;
A106:   H.(k+1) is real-valued by A98;
        H.(k+1) is E1-measurable by A1,A3,A55,A52,Th20,XBOOLE_1:36;
        hence thesis by A101,A105,A102,A106,A104,A103,MESFUNC2:11;
      end;
      hence thesis by A1,A3,A54,A55,A52,A65,Th20;
    end;
A107: E1 = dom(G.0) by A56,A65;
    for x be Element of X st x in E1 holds G#x is summable
    proof
      let x be Element of X;
      assume
A108: x in E1;
      E1 c= E by XBOOLE_1:36;
      then F#x is convergent by A6,A108;
      then H#x is convergent by A52,A108,Th12;
      then (Partial_Sums G)#x is convergent by A85,FUNCT_2:12;
      then Partial_Sums (G#x) is convergent by A107,A108,Th33;
      hence thesis;
    end;
    then consider J be ExtREAL_sequence such that
A109: for n be Nat holds J.n = Integral(M,(G.n)|E1) and
    J is summable and
A110: Integral(M,(lim(Partial_Sums G))|E1) = Sum J by A76,A107,A100,Th51;
    for n be object st n in NAT holds I.n = (Partial_Sums J).n
    proof
      let n be object;
      assume n in NAT;
      then reconsider n1 = n as Element of NAT;
A111: for n be Nat holds J.n = Integral(M,G.n)
      proof
        let n be Nat;
        dom(G.n) = E1 by A71;
        then (G.n)|E1 = (G.n);
        hence thesis by A109;
      end;
      E1 = dom(G.0) by A71;
      then (Partial_Sums J).n1 = Integral(M,(Partial_Sums G).n1) by A76,A100
,A111,Th46;
      then (Partial_Sums J).n1 = Integral(M,H.n1) by A85;
      then (Partial_Sums J).n1 = Integral(M,(F.n1)|E1) by A52;
      hence thesis by A53;
    end;
    then
A112: I = Partial_Sums J;
    dom(lim(Partial_Sums G)) = dom(H.0) by A97,MESFUNC8:def 9;
    then dom(lim(Partial_Sums G)) = E1 by A56;
    then
A113: lim I = Integral(M,lim H) by A97,A110,A112;
    take I;
A114: for x be Element of X st x in E1 holds F#x is convergent
    proof
      let x be Element of X;
A115: E1 c= E by XBOOLE_1:36;
      assume x in E1;
      hence thesis by A6,A115;
    end;
A116: for n be Element of NAT st 0 <= n holds (M*L).n = 0
    proof
      let n be Element of NAT;
      assume 0 <= n;
      dom L = NAT by FUNCT_2:def 1;
      then (M*L).n = M.(L.n) by FUNCT_1:13;
      then (M*L).n = M.(E /\ eq_dom(F.n,+infty)) by A50;
      hence thesis by A48;
    end;
    M*L is nonnegative by MEASURE2:1;
    then SUM(M*L) = Ser(M*L).0 by A116,SUPINF_2:48;
    then SUM(M*L) = (M*L).0 by SUPINF_2:def 11;
    then SUM(M*L) = 0 by A116;
    then M.(union E0) <= 0 by MEASURE2:11;
    then
A117: M.(union E0) = 0 by SUPINF_2:51;
A118: for n be Nat holds I.n = Integral(M,F.n)
    proof
      let n be Nat;
      reconsider n1=n as Element of NAT by ORDINAL1:def 12;
A119: I.n = Integral(M,(F.n1)|E1) by A53;
      dom(F.n) = E by A1,A3;
      hence thesis by A4,A117,A119,MESFUNC5:95;
    end;
    for n,m be Nat st m <= n holds I.m <= I.n
    proof
      let n,m be Nat;
A120: F.m is nonnegative by A8;
A121: dom(F.m) = E by A1,A3;
      assume m <= n;
      then
A122: for x be Element of X st x in dom(F.m) holds (F.m).x <= (F.n).x by A5
,A121;
A123: dom(F.n) = E by A1,A3;
A124: F.n is E-measurable by A4;
A125: F.n is nonnegative by A8;
      F.m is E-measurable by A4;
      then integral+(M,F.m) <= integral+(M,F.n) by A121,A123,A122,A120,A125
,A124,MESFUNC5:85;
      then Integral(M,F.m) <= integral+(M,F.n) by A4,A121,A120,MESFUNC5:88;
      then Integral(M,F.m) <= Integral(M,F.n) by A4,A123,A125,MESFUNC5:88;
      then I.m <= Integral(M,F.n) by A118;
      hence thesis by A118;
    end;
    then
A126: I is non-decreasing by RINFSUP2:7;
    E = dom(lim F) by A1,MESFUNC8:def 9;
    then
A127: Integral(M,lim F) = Integral(M,(lim F)|E1) by A7,A117,MESFUNC5:95;
    E1 c= dom(F.0) by A1,XBOOLE_1:36;
    hence thesis by A127,A52,A118,A126,A114,A113,Th19,RINFSUP2:37;
  end;
end;
