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;

theorem
  (for n be Nat holds F.n is E-measurable) implies (Partial_Sums F).m
  is E-measurable
proof
  assume
A1: for n be Nat holds F.n is E-measurable;
  then for n be Nat holds (Im F).n is E-measurable by Lm2;
  then (Partial_Sums Im F).m is E-measurable by Th16;
  then (Im(Partial_Sums F)).m is E-measurable by Th29;
  then
A2: Im((Partial_Sums F).m) is E-measurable by MESFUN7C:24;
  for n be Nat holds (Re F).n is E-measurable by A1,Lm2;
  then (Partial_Sums Re F).m is E-measurable by Th16;
  then (Re(Partial_Sums F)).m is E-measurable by Th29;
  then Re((Partial_Sums F).m) is E-measurable by MESFUN7C:24;
  hence (Partial_Sums F).m is E-measurable by A2,MESFUN6C:def 1;
end;
