reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;

theorem
  f is_integrable_on M & g is_integrable_on M implies dom (f+g) in S
proof
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
A3: Re g is_integrable_on M by A2;
A4: Im g is_integrable_on M by A2;
  Im f is_integrable_on M by A1;
  then dom(Im(f)+Im(g)) in S by A4,MESFUNC6:99;
  then
A5: dom(Im(f+g)) in S by Th5;
  Re f is_integrable_on M by A1;
  then dom(Re(f)+Re(g)) in S by A3,MESFUNC6:99;
  then
A6: dom(Re(f+g)) in S by Th5;
  dom(<i>(#)Im(f+g)) = dom(Im(f+g)) by VALUED_1:def 5;
  then dom(Re(f+g) + <i>(#)Im(f+g)) = dom(Re(f+g)) /\ dom(Im(f+g)) by
VALUED_1:def 1;
  then dom(Re(f+g) + <i>(#)Im(f+g)) in S by A6,A5,FINSUB_1:def 2;
  hence thesis by Th8;
end;
