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
  for c be Complex st f is_simple_func_in S holds c(#)f
  is_simple_func_in S
proof
  let c be Complex;
  assume
A1: f is_simple_func_in S;
  then
A2: Re f is_simple_func_in S by MESFUN7C:43;
  then
A3: (Im c)(#)(Re f) is_simple_func_in S by MESFUNC6:73;
A4: Im f is_simple_func_in S by A1,MESFUN7C:43;
  then (Im c)(#)(Im f) is_simple_func_in S by MESFUNC6:73;
  then (-1)(#)((Im c)(#)(Im f)) is_simple_func_in S by MESFUNC6:73;
  then
A5: R_EAL( -(Im c)(#)(Im f) ) is_simple_func_in S by MESFUNC6:49;
  (Re c)(#)(Re f) is_simple_func_in S by A2,MESFUNC6:73;
  then R_EAL( (Re c)(#)(Re f) ) is_simple_func_in S by MESFUNC6:49;
  then R_EAL( (Re c)(#)(Re f) ) + R_EAL( -(Im c)(#)(Im f) ) is_simple_func_in
  S by A5,MESFUNC5:38;
  then R_EAL((Re c)(#)(Re f) - (Im c)(#)(Im f)) is_simple_func_in S by
MESFUNC6:23;
  then (Re c)(#)(Re f) - (Im c)(#)(Im f) is_simple_func_in S by MESFUNC6:49;
  then
A6: Re(c(#)f) is_simple_func_in S by MESFUN6C:3;
  (Re c)(#)(Im f) is_simple_func_in S by A4,MESFUNC6:73;
  then (Im c)(#)(Re f) + (Re c)(#)(Im f) is_simple_func_in S by A3,MESFUNC6:72;
  then Im(c(#)f) is_simple_func_in S by MESFUN6C:3;
  hence thesis by A6,MESFUN7C:43;
end;
