
theorem Th95:
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 M1 is sigma_finite & E1 misses E2 holds
  Integral(M2,X-vol(E1 \/ E2,M1))
    = Integral(M2,X-vol(E1,M1)) + Integral(M2,X-vol(E2,M1))
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: M1 is sigma_finite and
A2: E1 misses E2;
   reconsider XX2 = X2 as Element of S2 by MEASURE1:7;
a3:X-vol(E1 \/ E2,M1) = X-vol(E1,M1) + X-vol(E2,M1) by A1,A2,Th93;
A3:dom(X-vol(E1,M1)) = XX2 & X-vol(E1,M1) is XX2-measurable
     by A1,DefXvol,FUNCT_2:def 1;
A4:dom(X-vol(E2,M1)) = XX2 & X-vol(E2,M1) is XX2-measurable
     by A1,DefXvol,FUNCT_2:def 1;
A5:dom(X-vol(E1 \/ E2,M1)) = XX2 & X-vol(E1 \/ E2,M1) is XX2-measurable
     by A1,DefXvol,FUNCT_2:def 1;
   reconsider V1 = X-vol(E1,M1) as PartFunc of X2,ExtREAL;
   reconsider V2 = X-vol(E2,M1) as PartFunc of X2,ExtREAL;
   ex Z be Element of S2 st
    Z = dom(X-vol(E1,M1) + X-vol(E2,M1)) &
    integral+(M2,X-vol(E1,M1) + X-vol(E2,M1))
      = integral+(M2,V1|Z) + integral+(M2,V2|Z)
        by A3,A4,MESFUNC5:78; then
   Integral(M2,X-vol(E1 \/ E2,M1))
     = integral+(M2,X-vol(E1,M1)) + integral+(M2,X-vol(E2,M1))
          by a3,A5,MESFUNC5:88
    .= Integral(M2,X-vol(E1,M1)) + integral+(M2,X-vol(E2,M1))
          by A3,MESFUNC5:88;
   hence thesis by A4,MESFUNC5:88;
end;
