
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 E1,E2 be Element of S, er be ExtReal holds
  Integral(M,chi(er,E1,X)|E2) = er * 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, er be ExtReal;
   reconsider XX = X as Element of S by MEASURE1:7;
   set f = chi(er,E1/\E2,X);
A1:chi(er,E1,X)|E2 = f|E2 by Th14;
A2:dom f = XX by FUNCT_2:def 1;
A3:E1 /\ E2 misses E2 \ E1 by XBOOLE_1:89;
A4:(E1 /\ E2) \/ (E2 \ E1) = E2 by XBOOLE_1:51;
   f is XX-measurable by Th13; then
X: ex W being Element of S st W = dom f & 
   f is W-measurable by A2;
   er >= 0 or er < 0; then
   f is nonnegative or f is nonpositive by Th17; then
A5:Integral(M,f|E2) = Integral(M,f|(E1/\E2)) + Integral(M,f|(E2\E1))
    by X,A3,A4,MESFUNC5:91,MESFUN11:62;
   dom(f|(E2\E1)) = dom f /\ (E2\E1) by RELAT_1:61; then
   dom(f|(E2\E1)) = X /\ (E2 \ E1) by FUNCT_2:def 1; then
A6:dom(f|(E2\E1)) = E2 \ E1 by XBOOLE_1:28;
   for x be object st x in dom(f|(E2\E1))
     holds (f|(E2\E1)).x >= 0 by Th16,XBOOLE_1:89; then
A7:f|(E2\E1) is nonnegative by SUPINF_2:52;
   for x be Element of X st x in dom(f|(E2\E1))
     holds (f|(E2\E1)).x = 0 by Th16,XBOOLE_1:89; then
   integral+(M,f|(E2\E1)) = 0 by A6,Th15,MESFUNC5:87; then
   Integral(M,f|(E2\E1)) = 0 by A6,A7,Th15,MESFUNC5:88; then
   Integral(M,f|E2) = er * M.(E1/\E2) + 0 by A5,Th50;
   hence Integral(M,chi(er,E1,X)|E2) = er * M.(E1/\E2) by A1,XXREAL_3:4;
end;
