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 Th23:
  z in dom((Partial_Sums F).n) & ((Partial_Sums F).n).z = +infty
  implies ex m be Nat st m <= n & (F.m).z = +infty
proof
  set PF = Partial_Sums F;
  assume that
A1: z in dom(PF.n) and
A2: (PF.n).z = +infty;
  now
    defpred P[Nat] means $1 <= n implies (PF.$1).z <> +infty;
    assume
A3: for m be Element of NAT st m <= n holds (F.m).z <> +infty;
A4: for k being Nat st P[k] holds P[k + 1]
    proof
      let k be Nat;
      assume
A5:   P[k];
      assume
A6:   k+1 <= n;
      then k <= n by NAT_1:13;
      then
A7:   z in dom(PF.k) by A1,Th22;
      not (PF.k).z in {+infty} by A5,A6,NAT_1:13,TARSKI:def 1;
      then not z in (PF.k)"{+infty} by FUNCT_1:def 7;
      then
A8:   not z in (PF.k)"{+infty} /\ (F.(k+1))"{-infty} by XBOOLE_0:def 4;
      z in dom(F.(k+1)) by A1,A6,Th22;
      then
A9:   z in dom(PF.k) /\ dom(F.(k+1)) by A7,XBOOLE_0:def 4;
A10:  PF.(k+1) = PF.k + F.(k+1) by Def4;
A11:  (F.(k+1)).z <> +infty by A3,A6;
      then not (F.(k+1)).z in {+infty} by TARSKI:def 1;
      then not z in (F.(k+1))"{+infty} by FUNCT_1:def 7;
      then not z in (PF.k)"{-infty} /\ (F.(k+1))"{+infty} by XBOOLE_0:def 4;
      then
      not z in ((PF.k)"{+infty} /\ (F.(k+1))"{-infty}) \/ ((PF.k)"{-infty
      } /\ (F.(k+1))"{+infty}) by A8,XBOOLE_0:def 3;
      then
      z in (dom(PF.k) /\ dom(F.(k+1))) \ (((PF.k)"{+infty} /\ (F.(k+1))"{
-infty}) \/ ((PF.k)"{-infty} /\ (F.(k+1))"{+infty})) by A9,XBOOLE_0:def 5;
      then z in dom(PF.(k+1)) by A10,MESFUNC1:def 3;
      then (PF.(k+1)).z = (PF.k).z + (F.(k+1)).z by A10,MESFUNC1:def 3;
      hence thesis by A5,A6,A11,NAT_1:13,XXREAL_3:16;
    end;
    PF.0 = F.0 by Def4;
    then
A12: P[ 0 ] by A3;
    for k being Nat holds P[k] from NAT_1:sch 2(A12,A4);
    hence contradiction by A2;
  end;
  hence thesis;
end;
