
theorem Th53:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,REAL, 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,REAL, A,B be set;
    assume that
A1:  A c= dom f and
A2:  B c= dom f and
A3:  f|A is_integrable_on M and
A4:  f|B is_integrable_on M;

A5: A c= dom(R_EAL f) & B c= dom(R_EAL f) by A1,A2,MESFUNC5:def 7;
    (R_EAL(f|A)) is_integrable_on M & (R_EAL(f|B)) is_integrable_on M
      by A3,A4,MESFUNC6:def 4; then
    (R_EAL f)|A is_integrable_on M & (R_EAL f)|B is_integrable_on M
      by Th16; then
    (R_EAL f)|(A\/B) is_integrable_on M by A5,Th52; then
    R_EAL(f|(A\/B)) is_integrable_on M by Th16;
    hence f|(A\/B) is_integrable_on M by MESFUNC6:def 4;
end;
