
theorem Th52:
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 set st
  A c= dom f & B c= dom f & f|A is_integrable_on M & f|B is_integrable_on M
  holds f|(A\/B) is_integrable_on M
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 set;
    assume that
A1:  A c= dom f & B c= dom f and
A2:  f|A is_integrable_on M and
A3:  f|B is_integrable_on M;

    set f1 = f|(A\/B);

A4: A = dom(f|A) & B = dom(f|B) by A1,RELAT_1:62;
A5: ex E be Element of S st E = dom(f|A) & f|A is E-measurable
      by A2,MESFUNC5:def 17; then
    reconsider A1=A as Element of S by A1,RELAT_1:62;

A6: ex E be Element of S st E = dom(f|B) & f|B is E-measurable
      by A3,MESFUNC5:def 17; then
    reconsider B1=B as Element of S by A1,RELAT_1:62;

A7: integral+(M,max+(f|A)) < +infty & integral+(M,max-(f|A)) < +infty
      by A2,MESFUNC5:def 17;

A8: integral+(M,max+(f|B)) < +infty & integral+(M,max-(f|B)) < +infty
      by A3,MESFUNC5:def 17;

A9: dom f1 = dom f /\ (A\/B) by RELAT_1:61
     .= (dom f /\ A) \/ (dom f /\ B) by XBOOLE_1:23;
A10: dom(f|A) = dom f /\ A & dom(f|B) = dom f /\ B by RELAT_1:61;

A11:f1|A = f|A & f1|B = f|B by XBOOLE_1:7,RELAT_1:74; then
    f1 is A1-measurable & f1 is B1-measurable by A4,A5,A6,MESFUN13:19; then
A12:f1 is (A1\/B1)-measurable by MESFUNC1:31;

A13:dom(max+f1) = dom f1 & dom(max-f1) = dom f1 by MESFUNC2:def 2,def 3;

A14:max+f1 is nonnegative & max-f1 is nonnegative by MESFUN11:5;

A15:(max+f1)|A = max+(f|A) & (max+f1)|B = max+(f|B) &
    (max-f1)|A = max-(f|A) & (max-f1)|B = max-(f|B) by A11,MESFUNC5:28;

A16:(max+f1)|(A\/B) = max+( (f|(A\/B))|(A\/B) ) by MESFUNC5:28
     .= max+(f|(A\/B));
A17:(max-f1)|(A\/B) = max-( (f|(A\/B))|(A\/B) ) by MESFUNC5:28
     .= max-(f|(A\/B));

A18:integral+(M,(max+f1)|(A\/B))
     <= integral+(M,(max+f1)|A) + integral+(M,(max+f1)|B)
       by A13,A4,A9,A10,A14,Th51,A12,MESFUNC2:25;
    integral+(M,(max+f1)|A) + integral+(M,(max+f1)|B) <> +infty
      by A7,A8,A15,XXREAL_3:16; then
A19:integral+(M,max+(f|(A\/B))) < +infty by A16,A18,XXREAL_0:2,4;

A20:integral+(M,(max-f1)|(A\/B))
     <= integral+(M,(max-f1)|A) + integral+(M,(max-f1)|B)
       by A13,A4,A9,A10,A14,Th51,A12,MESFUNC2:26;
    integral+(M,(max-f1)|A) + integral+(M,(max-f1)|B) <> +infty
      by A7,A8,A15,XXREAL_3:16; then
    integral+(M,max-(f|(A\/B))) < +infty by A17,A20,XXREAL_0:2,4;
    hence f|(A\/B) is_integrable_on M by A4,A9,A10,A12,A19,MESFUNC5:def 17;
end;
