
theorem Th93:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1,
  E1,E2 be Element of sigma measurable_rectangles(S1,S2)
 st M1 is sigma_finite & E1 misses E2 holds
  X-vol(E1 \/ E2,M1) = X-vol(E1,M1) + 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,
       E1,E2 be Element of sigma measurable_rectangles(S1,S2);
   assume that
A1: M1 is sigma_finite and
A2: E1 misses E2;
A3:dom(X-vol(E1 \/ E2,M1)) = X2 & dom(X-vol(E1,M1)) = X2
 & dom(X-vol(E2,M1)) = X2 by FUNCT_2:def 1; then
A4:dom(X-vol(E1,M1) + X-vol(E2,M1)) = X2 /\ X2 by MESFUNC5:22;
   for x be Element of X2 st x in dom(X-vol(E1 \/ E2,M1)) holds
      (X-vol(E1 \/ E2,M1)).x = (X-vol(E1,M1) + X-vol(E2,M1)).x
   proof
    let x be Element of X2;
    assume x in dom(X-vol(E1 \/ E2,M1));
A6: (X-vol(E1 \/ E2,M1)).x = M1.(Measurable-Y-section(E1 \/ E2,x))
  & (X-vol(E1,M1)).x = M1.(Measurable-Y-section(E1,x))
  & (X-vol(E2,M1)).x = M1.(Measurable-Y-section(E2,x)) by A1,DefXvol;

    Measurable-Y-section(E1 \/ E2,x)
     = Measurable-Y-section(E1,x) \/ Measurable-Y-section(E2,x) by Th20; then
    (X-vol(E1 \/ E2,M1)).x = (X-vol(E1,M1)).x + (X-vol(E2,M1)).x
      by A6,A2,Th29,MEASURE1:30;
    hence thesis by A4,MESFUNC1:def 3;
   end;
   hence thesis by A4,A3,PARTFUN1:5;
end;
