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 Th33:
  F is additive & F is with_the_same_dom & D c= dom(F.0) & x in D
implies ( Partial_Sums(F#x) is convergent_to_finite_number iff (Partial_Sums F)
  #x is convergent_to_finite_number ) & ( Partial_Sums(F#x) is
  convergent_to_+infty iff (Partial_Sums F)#x is convergent_to_+infty ) & (
  Partial_Sums(F#x) is convergent_to_-infty iff (Partial_Sums F)#x is
convergent_to_-infty ) & ( Partial_Sums(F#x) is convergent iff (Partial_Sums F)
  #x is convergent )
proof
  set PFx = Partial_Sums(F#x);
  set PF = Partial_Sums F;
  assume that
A1: F is additive and
A2: F is with_the_same_dom and
A3: D c= dom(F.0) and
A4: x in D;
  thus
A5: now
    assume PFx is convergent_to_finite_number;
    then consider g be Real such that
A6: for p be Real st 0<p ex n be Nat st for m be Nat st n<=m
    holds |. PFx.m -  g .| < p;
    now
      let p be Real;
      assume 0<p;
      then consider n be Nat such that
A7:   for m be Nat st n<=m holds |. PFx.m -  g .| < p by A6;
      take n;
      let m be Nat;
      assume
A8:   n<=m;
      PFx.m = (PF#x).m by A1,A2,A3,A4,Th32;
      hence |. (PF#x).m -  g .| < p by A7,A8;
    end;
    hence PF#x is convergent_to_finite_number;
  end;
  thus
A9: now
    assume PF#x is convergent_to_finite_number;
    then consider g be Real such that
A10: for p be Real st 0<p ex n be Nat st for m be Nat st n<=m
    holds |. (PF#x).m- g .| < p;
    now
      let p be Real;
      assume 0<p;
      then consider n be Nat such that
A11:  for m be Nat st n<=m holds |. (PF#x).m- g .| < p by A10;
      take n;
      let m be Nat;
      assume
A12:  n<=m;
      PFx.m = (PF#x).m by A1,A2,A3,A4,Th32;
      hence |. PFx.m -  g .| < p by A11,A12;
    end;
    hence PFx is convergent_to_finite_number;
  end;
  thus
A13: now
    assume
A14: PFx is convergent_to_+infty;
    now
      let r be Real;
      assume 0 < r;
      then consider n be Nat such that
A15:  for m be Nat st n <= m holds r <= PFx.m by A14;
      take n;
      let m be Nat;
      assume n <= m;
      then r <= PFx.m by A15;
      hence r <= (PF#x).m by A1,A2,A3,A4,Th32;
    end;
    hence PF#x is convergent_to_+infty;
  end;
  thus
A16: now
    assume
A17: PF#x is convergent_to_+infty;
    now
      let r be Real;
      assume 0 < r;
      then consider n be Nat such that
A18:  for m be Nat st n <= m holds r <= (PF#x).m by A17;
      take n;
      let m be Nat;
      assume n <= m;
      then r <= (PF#x).m by A18;
      hence r <= PFx.m by A1,A2,A3,A4,Th32;
    end;
    hence PFx is convergent_to_+infty;
  end;
  thus
A19: now
    assume
A20: PFx is convergent_to_-infty;
    now
      let r be Real;
      assume r < 0;
      then consider n be Nat such that
A21:  for m be Nat st n <= m holds PFx.m <= r by A20;
      take n;
      let m be Nat;
      assume n <= m;
      then PFx.m <= r by A21;
      hence (PF#x).m <= r by A1,A2,A3,A4,Th32;
    end;
    hence PF#x is convergent_to_-infty;
  end;
  thus
A22: now
    assume
A23: PF#x is convergent_to_-infty;
    now
      let r be Real;
      assume r < 0;
      then consider n be Nat such that
A24:  for m be Nat st n <= m holds (PF#x).m <= r by A23;
      take n;
      let m be Nat;
      assume n <= m;
      then (PF#x).m <= r by A24;
      hence PFx.m <= r by A1,A2,A3,A4,Th32;
    end;
    hence PFx is convergent_to_-infty;
  end;
  hereby
    assume
A25: PFx is convergent;
    per cases by A25;
    suppose
      PFx is convergent_to_+infty;
      hence PF#x is convergent by A13;
    end;
    suppose
      PFx is convergent_to_-infty;
      hence PF#x is convergent by A19;
    end;
    suppose
      PFx is convergent_to_finite_number;
      hence PF#x is convergent by A5;
    end;
  end;
  assume
A26: PF#x is convergent;
  per cases by A26;
  suppose
    PF#x is convergent_to_+infty;
    hence thesis by A16;
  end;
  suppose
    PF#x is convergent_to_-infty;
    hence thesis by A22;
  end;
  suppose
    PF#x is convergent_to_finite_number;
    hence thesis by A9;
  end;
end;
