reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th27:
  F is additive implies ((Partial_Sums F).n)"{-infty} /\ (F.(n+1))
  "{+infty} = {} & ((Partial_Sums F).n)"{+infty} /\ (F.(n+1))"{-infty} = {}
proof
  set PF = Partial_Sums F;
  assume
A1: F is additive;
  now
    given z be object such that
A2: z in (PF.n)"{-infty} and
A3: z in (F.(n+1))"{+infty};
A4: z in dom(PF.n) by A2,FUNCT_1:def 7;
    (PF.n).z in {-infty} by A2,FUNCT_1:def 7;
    then (PF.n).z = -infty by TARSKI:def 1;
    then consider k be Nat such that
A5: k <= n and
A6: (F.k).z = -infty by A4,Th25;
A7: z in dom(F.(n+1)) by A3,FUNCT_1:def 7;
    (F.(n+1)).z in {+infty} by A3,FUNCT_1:def 7;
    then
A8: (F.(n+1)).z = +infty by TARSKI:def 1;
    z in dom(F.k) by A4,A5,Th22;
    then z in dom(F.k) /\ dom(F.(n+1)) by A7,XBOOLE_0:def 4;
    hence contradiction by A1,A8,A6;
  end;
  then (PF.n)"{-infty} misses (F.(n+1))"{+infty} by XBOOLE_0:3;
  hence (PF.n)"{-infty} /\ (F.(n+1))"{+infty} = {} by XBOOLE_0:def 7;
  now
    given z be object such that
A9: z in (PF.n)"{+infty} and
A10: z in (F.(n+1))"{-infty};
A11: z in dom(PF.n) by A9,FUNCT_1:def 7;
    (PF.n).z in {+infty} by A9,FUNCT_1:def 7;
    then (PF.n).z = +infty by TARSKI:def 1;
    then consider k be Nat such that
A12: k <= n and
A13: (F.k).z = +infty by A11,Th23;
A14: z in dom(F.(n+1)) by A10,FUNCT_1:def 7;
    (F.(n+1)).z in {-infty} by A10,FUNCT_1:def 7;
    then
A15: (F.(n+1)).z = -infty by TARSKI:def 1;
    z in dom(F.k) by A11,A12,Th22;
    then z in dom(F.k) /\ dom(F.(n+1)) by A14,XBOOLE_0:def 4;
    hence contradiction by A1,A15,A13;
  end;
  then (PF.n)"{+infty} misses (F.(n+1))"{-infty} by XBOOLE_0:3;
  hence thesis by XBOOLE_0:def 7;
end;
