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 implies f|A is_simple_func_in S
proof
  assume
A1: f is_simple_func_in S;
  then Im f is_simple_func_in S by MESFUN7C:43;
  then R_EAL Im f is_simple_func_in S by MESFUNC6:49;
  then R_EAL((Im f)|A) is_simple_func_in S by MESFUNC5:34;
  then (Im f)|A is_simple_func_in S by MESFUNC6:49;
  then
A2: Im(f|A) is_simple_func_in S by MESFUN6C:7;
  Re f is_simple_func_in S by A1,MESFUN7C:43;
  then R_EAL Re f is_simple_func_in S by MESFUNC6:49;
  then R_EAL((Re f)|A) is_simple_func_in S by MESFUNC5:34;
  then (Re f)|A is_simple_func_in S by MESFUNC6:49;
  then Re(f|A) is_simple_func_in S by MESFUN6C:7;
  hence f|A is_simple_func_in S by A2,MESFUN7C:43;
end;
