 reserve Omega, Omega2 for non empty set;
 reserve Sigma, F for SigmaField of Omega;
 reserve Sigma2, F2 for SigmaField of Omega2;

theorem Th9:
for X being Function of Omega,REAL st
   X is (Sigma,Borel_Sets)-random_variable-like holds
(for k being Real holds
{w where w is Element of Omega: X.w >=k} is Element of Sigma &
{w where w is Element of Omega: X.w <k} is Element of Sigma) &
(for k1,k2 being Real st k1<k2 holds
{w where w is Element of Omega: k1 <= X.w & X.w < k2} is Element of Sigma) &
(for k1,k2 being Real st k1<=k2 holds
{w where w is Element of Omega: k1 <= X.w & X.w <= k2} is Element of Sigma) &
(for r being Real holds
       less_dom(X,r) = {w where w is Element of Omega: X.w <r}) &
(for r being Real holds
       great_eq_dom(X,r) = {w where w is Element of Omega: X.w >=r}) &
(for r being Real holds
       eq_dom(X,r) = {w where w is Element of Omega: X.w =r}) &
(for r being Real holds eq_dom(X,r) is Element of Sigma)
proof
 let X be Function of Omega,REAL;
 assume A1: X is (Sigma,Borel_Sets)-random_variable-like;
A2: for k being Real holds
 {w where w is Element of Omega: X.w >=k} is Element of Sigma &
 {w where w is Element of Omega: X.w <k} is Element of Sigma
proof
 let k be Real;
 A3: for q being set holds
 (ex w being Element of Omega st q=w & X.w >=k) iff
 (ex w being Element of Omega st q=w & X.w in [.k,+infty.])
 proof
 let q be set;
 now assume ex w being Element of Omega st q=w & X.w in [.k,+infty.]; then
  consider w being Element of Omega such that
  A4: q=w & X.w in [.k,+infty.];
  X.w in {z where z is Element of ExtREAL: k<=z & z<=+infty}
   by A4,XXREAL_1:def 1; then
  ex z being Element of ExtREAL st X.w=z & k<=z & z<=+infty;
  hence ex w being Element of Omega st q=w & X.w >=k by A4;
  end;
  hence thesis by XXREAL_1:219;
  end;
A5:for x being object holds
   x in {w where w is Element of Omega:X.w >=k} iff
   x in {w where w is Element of Omega:X.w in [.k,+infty.[}
  proof
   let x be object;
   x in {w where w is Element of Omega:X.w >=k} iff
   ex w being Element of Omega st x=w & X.w >=k; then
A6: x in {w where w is Element of Omega:X.w >=k} iff
   ex w being Element of Omega st x=w & X.w in [.k,+infty.] by A3;
   x in {w where w is Element of Omega:X.w in [.k,+infty.]} iff
   x in {w where w is Element of Omega:X.w in [.k,+infty.[}
   proof
    hereby assume
    x in {w where w is Element of Omega:X.w in [.k,+infty.]};
    then consider w being Element of Omega such that
    A7: w=x & X.w in [.k,+infty.];
    X.w in {a where a is Element of ExtREAL: k<=a & a<=+infty}
     by A7,XXREAL_1:def 1; then
    consider a being Element of ExtREAL such that
    A8: X.w = a & k<=a & a<=+infty;
    A9: X.w = a & k<=a & a<+infty by A8,XXREAL_0:9;
    {z where z is Element of ExtREAL: k<=z & z <+infty} =
      [.k,+infty.[ by XXREAL_1:def 2; then
    X.w in [.k,+infty.[ by A9;
    hence x in {g where g is Element of Omega:X.g in [.k,+infty.[} by A7;
    end;
    assume x in {w where w is Element of Omega:X.w in [.k,+infty.[}; then
    consider w being Element of Omega such that
     A10: w=x & X.w in [.k,+infty.[;
    w=x & X.w in {u where u is Element of ExtREAL: k<=u &
      u<+infty} by A10,XXREAL_1:def 2; then
    w=x & ex u being Element of ExtREAL st u=X.w & k<=u & u<+infty;
    then w=x & X.w in {u where u is Element of ExtREAL: k<=u & u<=+infty};
    then w=x & X.w in [.k,+infty.] by XXREAL_1:def 1;
    hence thesis;
   end;
   hence thesis by A6;
  end;
  k in REAL by XREAL_0:def 1; then
A11: [.k,+infty.[ is non empty by XXREAL_0:9,XXREAL_1:31;
A12: {w where w is Element of Omega:X.w >=k} =
  {w where w is Element of Omega:X.w is Element of [.k,+infty.[}
  proof
   {w where w is Element of Omega:X.w in [.k,+infty.[} =
   {w where w is Element of Omega:X.w is Element of [.k,+infty.[}
   proof
    for x being object holds
     x in {w where w is Element of Omega:X.w in [.k,+infty.[} iff
     x in {w where w is Element of Omega:X.w is Element of [.k,+infty.[}
    proof
     let x be object;
     hereby assume
       x in {w where w is Element of Omega:X.w in [.k,+infty.[}; then
       ex w being Element of Omega st w = x & X.w in [.k,+infty.[;
       hence
x in {w where w is Element of Omega:X.w is Element of [.k,+infty.[};
     end;
     assume
    x in {w where w is Element of Omega:X.w is Element of [.k,+infty.[};
    then consider w being Element of Omega such that
 A13: w = x & X.w is Element of [.k,+infty.[;
    thus thesis by A13,A11;
    end;
   hence thesis by TARSKI:2;
   end;
   hence thesis by A5,TARSKI:2;
 end;
A14: [.k,+infty.[ is Element of Borel_Sets &
    ].-infty,k.[ is Element of Borel_Sets by Th3;
A15: {w where w is Element of Omega: X.w is Element of [.k,+infty.[}
     = X"([.k,+infty.[) by Lm1,A11;
A16: for q being set holds
 (ex w being Element of Omega st q=w & X.w <k) iff
 (ex w being Element of Omega st q=w & X.w in [.-infty,k.[)
 proof
 let q be set;
  now assume ex w being Element of Omega st q=w & X.w in [.-infty,k.[; then
  consider w being Element of Omega such that
  A17: q=w & X.w in [.-infty,k.[;
  X.w in {z where z is Element of ExtREAL:-infty<=z & z<k}
    by A17,XXREAL_1:def 2; then
  ex z being Element of ExtREAL st X.w=z & -infty<=z & z<k;
  hence ex w being Element of Omega st q=w & X.w <k by A17;
  end;
  hence thesis by XXREAL_1:221;
  end;
   for x being object holds
   x in {w where w is Element of Omega:X.w <k} iff
   x in {w where w is Element of Omega:X.w in ].-infty,k.[}
  proof
   let x be object;
   x in {w where w is Element of Omega:X.w <k} iff
   ex w being Element of Omega st x=w & X.w <k; then
   A18: x in {w where w is Element of Omega:X.w <k} iff
   ex w being Element of Omega st x=w & X.w in [.-infty,k.[ by A16;
   x in {w where w is Element of Omega:X.w in [.-infty,k.[} iff
   x in {w where w is Element of Omega:X.w in ].-infty,k.[}
   proof
    hereby assume
    x in {w where w is Element of Omega:X.w in [.-infty,k.[}; then
    consider w being Element of Omega such that
A19: w=x & X.w in [.-infty,k.[;
    X.w in {a where a is Element of ExtREAL: -infty<=a & a<k}
     by A19,XXREAL_1:def 2; then
    consider a being Element of ExtREAL such that
    A20: X.w = a & -infty<=a & a<k;
    A21: X.w = a & -infty<a & a<k by A20,XXREAL_0:12;
    {z where z is Element of ExtREAL: -infty<z & z<k} =
       ].-infty,k.[ by XXREAL_1:def 4; then
    X.w in ].-infty,k.[ by A21;
    hence x in {g where g is Element of Omega:X.g in ].-infty,k.[} by A19;
    end;
    assume x in {w where w is Element of Omega:X.w in ].-infty,k.[}; then
    consider w being Element of Omega such that
    A22: w=x & X.w in ].-infty,k.[;
    w=x & X.w in {u where u is Element of ExtREAL: -infty<u & u<k}
      by A22,XXREAL_1:def 4; then
    w=x & ex u being Element of ExtREAL st u=X.w & -infty<u & u<k;
    then w=x &
    X.w in {u where u is Element of ExtREAL: -infty<=u & u<k}; then
    w=x & X.w in [.-infty,k.[ by XXREAL_1:def 2;
    hence thesis;
   end;
   hence thesis by A18;
  end; then
A23: {w where w is Element of Omega:X.w <k} =
  {w where w is Element of Omega:X.w in ].-infty,k.[} by TARSKI:2;
 {w where w is Element of Omega:X.w <k} is Element of Sigma
 proof
A24: [.k,+infty.[ is Element of Borel_Sets &
  ].-infty,k.[ is Element of Borel_Sets by Th3;
  k in REAL by XREAL_0:def 1; then
A25: ].-infty,k.[ is non empty by XXREAL_0:12,XXREAL_1:33; then
A26: {w where w is Element of Omega:X.w is Element of ].-infty,k.[}
     = X"(].-infty,k.[) by Lm1;
    for x being object holds
     x in {w where w is Element of Omega: X.w in ].-infty,k.[} iff
     x in {w where w is Element of Omega: X.w is Element of ].-infty,k.[}
    proof
     let x be object;
     hereby assume
       x in {w where w is Element of Omega:X.w in ].-infty,k.[}; then
       ex w being Element of Omega st w = x & X.w in ].-infty,k.[;
       hence
       x in {w where w is Element of Omega:X.w is Element of ].-infty,k.[};
     end;
     assume
     x in {w where w is Element of Omega:X.w is Element of ].-infty,k.[};
     then consider w being Element of Omega such that
 A27: w = x & X.w is Element of ].-infty,k.[;
     thus thesis by A27,A25;
    end; then
  {w where w is Element of Omega:X.w <k} =
  {w where w is Element of Omega:X.w is Element of ].-infty,k.[}
    by A23,TARSKI:2
    .= X"(].-infty,k.[) by A26;
  hence thesis by A24,A1;
 end;
hence thesis by A14,A1,A12,A15;
end;

A28: for k1,k2 being Real st k1<k2 holds
{w where w is Element of Omega: k1 <= X.w & X.w < k2} is Element of Sigma
proof
 let k1,k2 be Real;
 assume A29: k1<k2; then
 A30: [.k1,k2.[ is non empty by XXREAL_1:31;
 {w where w is Element of Omega:k1 <= X.w & X.w < k2} is Element of Sigma
 proof
   for x being object holds
   x in {w where w is Element of Omega:k1 <= X.w & X.w < k2} iff
   x in {w where w is Element of Omega:X.w is Element of [.k1,k2.[}
  proof
   let x be object;
   hereby assume x in {w where w is Element of Omega:k1 <= X.w & X.w < k2};
      then consider w being Element of Omega such that
 A31: x=w & k1 <= X.w & X.w < k2;
      reconsider a=X.w as Element of ExtREAL by XXREAL_0:def 1;
      a=X.w; then
      X.w in {z where z is Element of ExtREAL: k1 <= z & z < k2} by A31; then
      X.w is Element of [.k1,k2.[ by XXREAL_1:def 2;
      hence x in {w1 where w1 is Element of Omega:
           X.w1 is Element of [.k1,k2.[} by A31;
      end;
      assume x in {w where w is Element of Omega:
        X.w is Element of [.k1,k2.[}; then
      consider w being Element of Omega such that
     A32: x=w & X.w is Element of [.k1,k2.[;
      A33: [.k1,k2.[ is non empty by A29,XXREAL_1:31;
      X.w in [.k1,k2.[ by A32,A33; then
      X.w in {a where a is Element of ExtREAL:
              k1 <=a & a <k2} by XXREAL_1:def 2; then
      ex a being Element of ExtREAL st a=X.w & k1 <=a & a <k2;
      hence thesis by A32;
  end; then
  A34: {w where w is Element of Omega:k1 <= X.w & X.w < k2} =
  {w where w is Element of Omega:X.w is Element of [.k1,k2.[} by TARSKI:2;
A35:  [.k2,k1.[ is Element of Borel_Sets &
  [.k1,k2.[ is Element of Borel_Sets by Th4;
  {w where w is Element of Omega: X.w is Element of [.k1,k2.[}
   = X"([.k1,k2.[) by Lm1,A30;
  hence thesis by A1,A34,A35;
end;
hence thesis;
end;
A36: for k1,k2 being Real st k1<=k2 holds
  {w where w is Element of Omega: k1 <= X.w & X.w <= k2} is Element of Sigma
proof
  let k1,k2 be Real;
  assume A37: k1<=k2; then
A38: [.k1,k2.] is non empty by XXREAL_1:30;
  for x being object holds
   (x in {w where w is Element of Omega:k1 <= X.w & X.w <= k2} iff
   x in {w where w is Element of Omega:X.w is Element of [.k1,k2.]})
  proof
   let x be object;
   hereby assume x in {w where w is Element of Omega:k1 <= X.w & X.w <= k2};
      then consider w being Element of Omega such that
A39:  x=w & k1 <= X.w & X.w <= k2;
      reconsider a=X.w as Element of ExtREAL by XXREAL_0:def 1;
      a=X.w; then
      X.w in {z where z is Element of ExtREAL:
         k1 <= z & z <= k2} by A39; then
      X.w is Element of [.k1,k2.] by XXREAL_1:def 1;
      hence x in {w1 where w1 is Element of Omega:
         X.w1 is Element of [.k1,k2.]} by A39;
      end;
      assume x in {w where w is Element of Omega:
        X.w is Element of [.k1,k2.]}; then
      consider w being Element of Omega such that
       A40: x=w & X.w is Element of [.k1,k2.];
      A41: [.k1,k2.] is non empty by A37,XXREAL_1:30;
      X.w in [.k1,k2.] by A40,A41; then
      X.w in {a where a is Element of ExtREAL:
              k1 <=a & a <=k2} by XXREAL_1:def 1; then
      ex a being Element of ExtREAL st a=X.w & k1 <=a & a <=k2;
      hence thesis by A40;
  end; then
  A42: {w where w is Element of Omega:k1 <= X.w & X.w <= k2} =
  {w where w is Element of Omega:X.w is Element of [.k1,k2.]} by TARSKI:2;
A43: [.k1,k2.[ is Element of Borel_Sets &
  [.k1,k2.] is Element of Borel_Sets by Th8,Th4;
  {w where w is Element of Omega: X.w is Element of [.k1,k2.]}
    = X"([.k1,k2.]) by Lm1,A38;
  hence thesis by A1,A42,A43;
end;
A44: for r being Real holds
       less_dom(X,r) = {w where w is Element of Omega: X.w <r}
  proof
   let r be Real;
   for x being object holds
    x in less_dom(X,r) iff x in {w where w is Element of Omega:X.w <r}
   proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
     x in less_dom(X,r) iff x in dom X & X.xx < r by MESFUNC1:def 11; then
    x in less_dom(X,r) iff x in Omega & X.xx < r by FUNCT_2:def 1; then
    x in less_dom(X,r) iff ex q being Element of Omega st q=x & X.q<r;
    hence thesis;
   end;
  hence thesis by TARSKI:2;
end;

A45: for r being Real holds
  great_eq_dom(X,r) = {w where w is Element of Omega: X.w >=r}
  proof
   let r be Real;
   for x being object holds x in great_eq_dom(X,r) iff
    x in {w where w is Element of Omega:X.w >=r}
   proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    x in great_eq_dom(X,r) iff x in dom X & X.xx >= r by MESFUNC1:def 14; then
    x in great_eq_dom(X,r) iff x in Omega & X.xx >= r by FUNCT_2:def 1; then
    x in great_eq_dom(X,r) iff ex q being Element of Omega st q=x & X.q>=r;
    hence thesis;
   end;
   hence thesis by TARSKI:2;
end;

A46: for r being Real holds
       eq_dom(X,r) = {w where w is Element of Omega: X.w = r}
  proof
   let r be Real;
   for x being object holds x in eq_dom(X,r) iff
    x in {w where w is Element of Omega:X.w =r}
   proof
    let x be object;
    x in eq_dom(X,r) iff x in dom X & X.x = r by MESFUNC1:def 15; then
    x in eq_dom(X,r) iff x in Omega & X.x = r by FUNCT_2:def 1; then
    x in eq_dom(X,r) iff (ex q being Element of Omega st q=x & X.q=r);
    hence thesis;
   end;
   hence thesis by TARSKI:2;
end;
  for r being Real holds eq_dom(X,r) is Element of Sigma
  proof
    let r be Real;
    for x being object holds
      x in {w where w is Element of Omega: r <= X.w & X.w <= r} iff
       x in {w where w is Element of Omega: X.w=r}
    proof
      let x be object;
      x in {w where w is Element of Omega: r <= X.w & X.w <= r} iff
      ex q being Element of Omega st x=q & r<=X.q & X.q<=r; then
      x in {w where w is Element of Omega: r <= X.w & X.w <= r} iff
      ex q being Element of Omega st x=q & X.q=r by XXREAL_0:1;
      hence thesis;
    end; then
    {w where w is Element of Omega: r <= X.w & X.w <= r} =
    {w where w is Element of Omega: X.w=r} by TARSKI:2; then
    {w where w is Element of Omega: X.w=r} is Element of Sigma by A36;
    hence thesis by A46;
  end;
  hence thesis by A2,A28,A36,A44,A45,A46;
end;
