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
  F is with_the_same_dom & E c= dom(F.0) & (for m be Nat holds F.m
  is E-measurable) implies (F||E).n is E-measurable
proof
  set G = F||E;
A1: Re G = (Re F)||E by Th21;
A2: Im G = (Im F)||E by Th22;
  assume F is with_the_same_dom;
  then
A3: Re F is with_the_same_dom;
  then
A4: Im F is with_the_same_dom by Th25;
  assume
A5: E c= dom(F.0);
  assume
A6: for m be Nat holds F.m is E-measurable;
A7: for m be Nat holds (Re F).m is E-measurable & (Im F).m
  is E-measurable
  proof
    let m be Nat;
    F.m is E-measurable by A6;
    then Re(F.m) is E-measurable & Im(F.m) is E-measurable by
MESFUN6C:def 1;
    hence thesis by MESFUN7C:24;
  end;
  E c= dom((Im F).0) by A5,MESFUN7C:def 12;
  then (Im G).n is E-measurable by A4,A2,A7,Th4;
  then
A8: Im(G.n) is E-measurable by MESFUN7C:24;
  E c= dom((Re F).0) by A5,MESFUN7C:def 11;
  then (Re G).n is E-measurable by A3,A1,A7,Th4;
  then Re(G.n) is E-measurable by MESFUN7C:24;
  hence thesis by A8,MESFUN6C:def 1;
end;
