
theorem Th94:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
  E1,E2 be Element of sigma measurable_rectangles(S1,S2)
 st M2 is sigma_finite & E1 misses E2 holds
  Integral(M1,Y-vol(E1 \/ E2,M2))
    = Integral(M1,Y-vol(E1,M2)) + Integral(M1,Y-vol(E2,M2))
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
       M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
       E1,E2 be Element of sigma measurable_rectangles(S1,S2);
   assume that
A1: M2 is sigma_finite and
A2: E1 misses E2;
   reconsider XX1 = X1 as Element of S1 by MEASURE1:7;
a3:Y-vol(E1 \/ E2,M2) = Y-vol(E1,M2) + Y-vol(E2,M2) by A1,A2,Th92;
A3:dom(Y-vol(E1,M2)) = XX1 & Y-vol(E1,M2) is XX1-measurable
     by A1,DefYvol,FUNCT_2:def 1;
A4:dom(Y-vol(E2,M2)) = XX1 & Y-vol(E2,M2) is XX1-measurable
     by A1,DefYvol,FUNCT_2:def 1;
A5:dom(Y-vol(E1 \/ E2,M2)) = XX1 & Y-vol(E1 \/ E2,M2) is XX1-measurable
     by A1,DefYvol,FUNCT_2:def 1;
   reconsider Y1 = Y-vol(E1,M2) as PartFunc of X1,ExtREAL;
   reconsider Y2 = Y-vol(E2,M2) as PartFunc of X1,ExtREAL;
   ex Z be Element of S1 st
    Z = dom(Y-vol(E1,M2) + Y-vol(E2,M2)) &
    integral+(M1,Y-vol(E1,M2) + Y-vol(E2,M2))
      = integral+(M1,Y1|Z) + integral+(M1,Y2|Z)
        by A3,A4,MESFUNC5:78; then
   Integral(M1,Y-vol(E1 \/ E2,M2))
     = integral+(M1,Y-vol(E1,M2)) + integral+(M1,Y-vol(E2,M2))
          by a3,A5,MESFUNC5:88
    .= Integral(M1,Y-vol(E1,M2)) + integral+(M1,Y-vol(E2,M2))
          by A3,MESFUNC5:88;
   hence thesis by A4,MESFUNC5:88;
end;
