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
  E c= dom(F.0) & F is with_the_same_dom & (for x be Element of X st x
in E holds F#x is summable) implies for x be Element of X st x in E holds (F||E
  )#x is summable
proof
  set G = F||E;
  assume that
A1: E c= dom(F.0) and
A2: F is with_the_same_dom and
A3: for x be Element of X st x in E holds F#x is summable;
A4: G is with_the_same_dom by A2,Th2;
A5: for x be Element of X st x in E holds (Im F)#x is summable
  proof
    let x be Element of X;
    assume
A6: x in E;
    then F#x is summable by A3;
    then Im(F#x) is summable;
    hence (Im F)#x is summable by A1,A2,A6,MESFUN7C:23;
  end;
A7: for x be Element of X st x in E holds (Re F)#x is summable
  proof
    let x be Element of X;
    assume
A8: x in E;
    then F#x is summable by A3;
    then Re(F#x) is summable;
    hence (Re F)#x is summable by A1,A2,A8,MESFUN7C:23;
  end;
  hereby
    let x be Element of X;
    assume
A9: x in E;
    G.0= (F.0)|E by Def1;
    then
A10: x in dom(G.0) by A1,A9,RELAT_1:62;
    Im G = (Im F)||E by Th22;
    then (Im G)#x is summable by A5,A9,Th6;
    then
A11: Im(G#x) is summable by A4,A10,MESFUN7C:23;
    Re G = (Re F)||E by Th21;
    then (Re G)#x is summable by A7,A9,Th6;
    then Re(G#x) is summable by A4,A10,MESFUN7C:23;
    hence G#x is summable by A11,COMSEQ_3:57;
  end;
end;
