reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;
reserve F for Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX,
  A for set;
reserve f,g for PartFunc of X,COMPLEX,
  A for Element of S;

theorem
  f is_simple_func_in S & g is_simple_func_in S implies f+g is_simple_func_in S
proof
  assume
A1: f is_simple_func_in S & g is_simple_func_in S;
  then Im f is_simple_func_in S & Im g is_simple_func_in S by MESFUN7C:43;
  then Im f + Im g is_simple_func_in S by MESFUNC6:72;
  then
A2: Im(f+g) is_simple_func_in S by MESFUN6C:5;
  Re f is_simple_func_in S & Re g is_simple_func_in S by A1,MESFUN7C:43;
  then Re f + Re g is_simple_func_in S by MESFUNC6:72;
  then Re(f+g) is_simple_func_in S by MESFUN6C:5;
  hence thesis by A2,MESFUN7C:43;
end;
