
theorem Th20:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 E1,E2 be Element of S holds Integral(M,(chi(E1,X))|E2) = M.(E1/\ E2)
proof
   let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
   E1,E2 be Element of S;
   reconsider XX = X as Element of S by MEASURE1:7;
A1:E2 = (E1 /\ E2) \/ (E2 \ E1) by XBOOLE_1:51;
   set F = E2\E1;
A2:dom((chi(E1,X))|(E1/\E2)) = dom(chi(E1,X)) /\ (E1/\E2) by RELAT_1:61
    .= X /\ (E1/\E2) by FUNCT_3:def 3;
A3:dom(chi(E1/\E2,X)|(E1/\E2)) = dom(chi(E1/\E2,X)) /\ (E1/\E2) by RELAT_1:61
    .= X /\ (E1/\E2) by FUNCT_3:def 3;
   now
    let x be Element of X;
    assume
A4: x in dom((chi(E1,X))|(E1/\E2)); then
A5: (chi(E1/\E2,X)|(E1/\E2)).x = (chi(E1/\E2,X)).x by A2,A3,FUNCT_1:47;
A6: x in E1 /\ E2 by A2,A4,XBOOLE_0:def 4; then
A7: x in E1 by XBOOLE_0:def 4;
    ((chi(E1,X))|(E1/\E2)).x = (chi(E1,X)).x by A4,FUNCT_1:47
      .= 1 by A7,FUNCT_3:def 3;
    hence ((chi(E1,X))|(E1/\E2)).x = (chi(E1/\E2,X)|(E1/\E2)).x
      by A6,A5,FUNCT_3:def 3;
   end; then
   (chi(E1,X))|(E1/\E2) = chi(E1/\E2,X)|(E1/\E2) by A2,A3,PARTFUN1:5; then
A9:Integral(M,(chi(E1,X))|(E1/\E2)) = M.(E1/\E2) by MESFUNC9:14;
A10:XX = dom chi(E1,X) by FUNCT_3:def 3; then
A11:F = dom((chi(E1,X))|(E2\E1)) by RELAT_1:62; then
   F = dom(chi(E1,X)) /\ F by RELAT_1:61; then
A12: (chi(E1,X))|(E2\E1) is F-measurable by MESFUNC2:29,MESFUNC5:42;

   now
    let x be Element of X;
    assume
A15: x in dom ((chi(E1,X))|(E2\E1));
    E2 \ E1 c= X \ E1 by XBOOLE_1:33;
    then (chi(E1,X)).x = 0 by A11,A15,FUNCT_3:37;
    hence 0= ((chi(E1,X))|(E2\E1)).x by A15,FUNCT_1:47;
   end;
   then integral+(M,(chi(E1,X))|(E2\E1)) = 0 by A11,A12,MESFUNC5:87;
   then
A16: Integral(M,(chi(E1,X))|(E2\E1)) = 0. by A11,A12,MESFUNC5:15,88;
   chi(E1,X) is XX-measurable by MESFUNC2:29;
   then Integral(M,(chi(E1,X))|E2) = Integral(M,(chi(E1,X))|(E1/\E2)) +
   Integral(M,(chi(E1,X))|(E2\E1)) by A10,A1,MESFUNC5:91,XBOOLE_1:89;
   hence thesis by A9,A16,XXREAL_3:4;
end;
