reserve Omega, I for non empty set;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve D, E, F for Subset-Family of Omega;
reserve  B, sB for non empty Subset of Sigma;
reserve b for Element of B;
reserve a for Element of Sigma;
reserve p, q, u, v for Event of Sigma;
reserve n, m for Element of NAT;
reserve S, S9, X, x, y, z, i, j for set;

theorem Th11:
  for F being ManySortedSigmaField of I, Sigma, J being non empty
  Subset of I holds sigma MeetSections(J,F) = sigUn(F,J)
proof
  let F be ManySortedSigmaField of I,Sigma, J be non empty Subset of I;
A1: Union (F|J) c= MeetSections(J,F)
  proof
    let x be object;
    assume x in Union (F|J);
    then consider y such that
A2: x in y and
A3: y in rng (F|J) by TARSKI:def 4;
    consider i being object such that
A4: i in dom (F|J) and
A5: y = (F|J).i by A3,FUNCT_1:def 3;
    reconsider i as set by TARSKI:1;
    y = (F|J).i by A5;
    then reconsider x as Subset of Omega by A2;
    defpred P[object,object] means $2=x & $2 in F.$1;
A6: {i} c= J by A4,ZFMISC_1:31;
    then reconsider E={i} as finite Subset of I by XBOOLE_1:1;
A7: for j being object st j in E ex y being object st y in Sigma & P[j,y]
    proof
      let j be object;
      assume
A8:   j in E;
      i in I by A4,TARSKI:def 3;
      then
A9:   F.i in bool Sigma by FUNCT_2:5;
      take y=x;
      y in F.i by A2,A4,A5,FUNCT_1:49;
      hence thesis by A8,A9,TARSKI:def 1;
    end;
    consider g being Function of E, Sigma such that
A10: for j being object st j in E holds P[j,g.j] from FUNCT_2:sch 1(A7);
    for i st i in E holds g.i in F.i by A10;
    then reconsider g as SigmaSection of E,F by Def4;
    dom g = E by FUNCT_2:def 1;
    then
A11: rng g = {g.i} by FUNCT_1:4;
    i in E by TARSKI:def 1;
    then x=g.i by A10
      .= meet rng g by A11,SETFAM_1:10;
    hence thesis by A6,Def9;
  end;
  MeetSections(J,F) c= sigma Union (F|J)
  proof
    let x be object;
    assume x in MeetSections(J,F);
    then consider
    E being non empty finite Subset of I, f being SigmaSection of E,F
    such that
A12: E c= J and
A13: x = meet rng f by Def9;
    reconsider Ee=E as Element of Fin E by FINSUB_1:def 5;
    for B being Element of Fin E holds meet rng (f|B) in sigma (Union (F|J ))
    proof
      defpred P[set] means meet rng (f|$1) in sigma Union (F|J);
      let B be Element of Fin E;
A14:  for B9 being Element of Fin E, b being Element of E holds P[B9] &
      not b in B9 implies P[B9 \/ {b}]
      proof
        let B9 be Element of Fin E, b be Element of E;
        assume that
A15:    meet rng (f|B9) in sigma (Union (F|J)) and
        not b in B9;
        reconsider rfb = rng (f|{b}) as set;
        reconsider rfB9 = rng (f|B9) as set;
        reconsider rfB9b = rng (f|(B9 \/ {b})) as set;
        rng (f|(B9 \/ {b})) = rng ((f|B9) \/ (f|{b})) by RELAT_1:78;
        then
A16:    rng (f|(B9 \/ {b})) = (rng (f|B9)) \/ rng (f|{b}) by RELAT_1:12;
        dom (F|J) = J by FUNCT_2:def 1;
        then
A17:    b in dom (F|J) by A12;
        then (F|J).b in rng (F|J) by FUNCT_1:def 3;
        then F.b in rng (F|J) by A17,FUNCT_1:47;
        then
A18:    F.b c= Union (F|J) by ZFMISC_1:74;
        Union (F|J) c= sigma Union (F|J) by PROB_1:def 9;
        then
A19:    F.b c= sigma (Union (F|J)) by A18;
        b is Element of dom f by FUNCT_2:def 1;
        then
A20:    {b} = dom f /\ {b} by ZFMISC_1:46
          .= dom (f|{b}) by RELAT_1:61;
        then
A21:    b in dom (f|{b}) by TARSKI:def 1;
        rng (f|{b}) = {(f|{b}).b} by A20,FUNCT_1:4
          .= {f.b} by A21,FUNCT_1:47;
        then meet rng (f|{b}) = f.b by SETFAM_1:10;
        then
A22:    meet rng (f|{b}) in F.b by Def4;
        per cases;
        suppose
          rng (f|B9) = {};
          hence thesis by A22,A19,A16;
        end;
        suppose
A23:      not rng (f|B9) = {};
          dom f = E & b in {b} by FUNCT_2:def 1,TARSKI:def 1;
          then rfb <> {} by FUNCT_1:50;
          then meet rfB9b = (meet rfB9) /\ meet rfb by A16,A23,SETFAM_1:9;
          then meet rng (f|(B9 \/ {b})) is Event of sigma Union (F|J) by A15
,A22,A19,PROB_1:19;
          hence thesis;
        end;
      end;
      meet rng (f|{}) = {} by SETFAM_1:def 1;
      then
A24:  P[{}.E] by PROB_1:4;
      for B1 being Element of Fin E holds P[B1] from SETWISEO:sch 2( A24,
      A14);
      hence thesis;
    end;
    then meet rng (f|Ee) in sigma (Union (F|J));
    hence thesis by A13;
  end;
  hence sigma MeetSections(J,F) c= sigUn(F,J) by PROB_1:def 9;
  MeetSections(J,F) c= sigma MeetSections(J,F) by PROB_1:def 9;
  then Union (F|J) c= sigma MeetSections(J,F) by A1;
  hence thesis by PROB_1:def 9;
end;
