
theorem Th92:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  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
  Y-vol(E1 \/ E2,M2) = Y-vol(E1,M2) + Y-vol(E2,M2)
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
       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;
A3:dom(Y-vol(E1 \/ E2,M2)) = X1 & dom(Y-vol(E1,M2)) = X1
 & dom(Y-vol(E2,M2)) = X1 by FUNCT_2:def 1; then
A4:dom(Y-vol(E1,M2) + Y-vol(E2,M2)) = X1 /\ X1 by MESFUNC5:22;
   for x be Element of X1 st x in dom(Y-vol(E1 \/ E2,M2)) holds
      (Y-vol(E1 \/ E2,M2)).x = (Y-vol(E1,M2) + Y-vol(E2,M2)).x
   proof
    let x be Element of X1;
    assume x in dom(Y-vol(E1 \/ E2,M2));
A6: (Y-vol(E1 \/ E2,M2)).x = M2.(Measurable-X-section(E1 \/ E2,x))
  & (Y-vol(E1,M2)).x = M2.(Measurable-X-section(E1,x))
  & (Y-vol(E2,M2)).x = M2.(Measurable-X-section(E2,x)) by A1,DefYvol;
    Measurable-X-section(E1 \/ E2,x)
     = Measurable-X-section(E1,x) \/ Measurable-X-section(E2,x) by Th20; then
    (Y-vol(E1 \/ E2,M2)).x = (Y-vol(E1,M2)).x + (Y-vol(E2,M2)).x
      by A6,A2,Th29,MEASURE1:30;
    hence thesis by A4,MESFUNC1:def 3;
   end;
   hence thesis by A4,A3,PARTFUN1:5;
end;
