
theorem Th98:
  for X be non empty set, S be SigmaField of X, M be
  sigma_Measure of S, f be PartFunc of X,ExtREAL, A,B be Element of S st f
is_integrable_on M & A misses B holds Integral(M,f|(A\/B)) = Integral(M,f|A) +
  Integral(M,f|B)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, A,B be Element of S;
  assume that
A1: f is_integrable_on M and
A2: A misses B;
  consider E be Element of S such that
A3: E = dom f and
A4: f is E-measurable by A1;
  set AB = E/\(A\/B);
A5: max+(f|A)=max+f|A by Th28;
A6: dom f = dom(max-f) by MESFUNC2:def 3;
  then max-f|(A\/B) = max-f|E|(A\/B) by A3,GRFUNC_1:23;
  then
A7: max-f|(A\/B) = max-f|AB by RELAT_1:71;
  max-f is nonnegative by Lm1;
  then
A8: integral+(M,max-f|(A\/B)) = integral+(M,max-f|A) + integral+(M,max-f|B)
  by A2,A3,A4,A6,Th81,MESFUNC2:26;
A9: f|A is_integrable_on M by A1,Th97;
  then
A10: 0 <= integral+(M,max+(f|A)) by Th96;
A11: f|B is_integrable_on M by A1,Th97;
  then
A12: 0 <= integral+(M,max+(f|B)) by Th96;
A13: 0 <= integral+(M,max-(f|B)) by A11,Th96;
  integral+(M,max-(f|B)) < +infty by A11;
  then reconsider g2 = integral+(M,max-(f|B)) as Element of REAL
     by A13,XXREAL_0:14;
  integral+(M,max+(f|B)) < +infty by A11;
  then reconsider g1 = integral+(M,max+(f|B)) as Element of REAL
     by A12,XXREAL_0:14;
A14: integral+(M,max+(f|B))-integral+(M,max-(f|B)) = g1-g2 by SUPINF_2:3;
A15: max-(f|A)=max-f|A by Th28;
A16: dom f= dom(max+f) by MESFUNC2:def 2;
  then max+f|(A\/B) = max+f|E|(A\/B) by A3,GRFUNC_1:23;
  then
A17: max+f|(A\/B) = max+f|AB by RELAT_1:71;
  max+f is nonnegative by Lm1;
  then
A18: integral+(M,max+f|(A\/B)) = integral+(M,max+f|A) + integral+(M,max+f |B
  ) by A2,A3,A4,A16,Th81,MESFUNC2:25;
A19: max-(f|B)=max-f|B by Th28;
A20: max+(f|B)=max+f|B by Th28;
  integral+(M,max+(f|A)) < +infty by A9;
  then reconsider f1 = integral+(M,max+(f|A)) as Element of REAL
    by A10,XXREAL_0:14;
A21: integral+(M,max+(f|A)) + integral+(M,max+(f|B)) = f1+g1 by SUPINF_2:1;
A22: 0 <= integral+(M,max-(f|A)) by A9,Th96;
  integral+(M,max-(f|A)) < +infty by A9;
  then reconsider f2 = integral+(M,max-(f|A)) as Element of REAL
    by A22,XXREAL_0:14;
A23: integral+(M,max-(f|A)) + integral+(M,max-(f|B)) = f2+g2 by SUPINF_2:1;
  Integral(M,f|(A\/B)) = Integral(M,(f|E)|(A\/B)) by A3,GRFUNC_1:23
    .= Integral(M,f|AB) by RELAT_1:71
    .= integral+(M,max+f|AB) - integral+(M,max-(f|AB)) by Th28
    .= integral+(M,max+f|AB) - integral+(M,max-f|AB) by Th28;
  then Integral(M,f|(A\/B)) = f1+g1-(f2+g2) by A18,A8,A17,A7,A5,A15,A20,A19,A21
,A23,SUPINF_2:3;
  then
A24: Integral(M,f|(A\/B)) = (f1-f2)+(g1-g2);
  integral+(M,max+(f|A))-integral+(M,max-(f|A)) = f1-f2 by SUPINF_2:3;
  hence thesis by A24,A14,SUPINF_2:1;
end;
