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_integrable_on M) implies for m be Nat holds
  (Partial_Sums F).m is_integrable_on M
proof
  assume
A1: for n be Nat holds F.n is_integrable_on M;
A2: for n be Nat holds (Im F).n is_integrable_on M
  proof
    let n be Nat;
    F.n is_integrable_on M by A1;
    then Im(F.n) is_integrable_on M by MESFUN6C:def 2;
    hence (Im F).n is_integrable_on M by MESFUN7C:24;
  end;
A3: for n be Nat holds (Re F).n is_integrable_on M
  proof
    let n be Nat;
    F.n is_integrable_on M by A1;
    then Re(F.n) is_integrable_on M by MESFUN6C:def 2;
    hence (Re F).n is_integrable_on M by MESFUN7C:24;
  end;
  thus for m be Nat holds (Partial_Sums F).m is_integrable_on M
  proof
    let m be Nat;
    (Partial_Sums Im F).m is_integrable_on M by A2,Th19;
    then (Im(Partial_Sums F)).m is_integrable_on M by Th29;
    then
A4: Im((Partial_Sums F).m) is_integrable_on M by MESFUN7C:24;
    (Partial_Sums Re F).m is_integrable_on M by A3,Th19;
    then (Re(Partial_Sums F)).m is_integrable_on M by Th29;
    then Re((Partial_Sums F).m) is_integrable_on M by MESFUN7C:24;
    hence (Partial_Sums F).m is_integrable_on M by A4,MESFUN6C:def 2;
  end;
end;
