theorem
  F is additive & z in dom((Partial_Sums F).n) & ((Partial_Sums F).n).z
  = +infty & m <= n implies (F.m).z <> -infty
proof
  assume that
A1: F is additive and
A2: z in dom((Partial_Sums F).n) and
A3: ((Partial_Sums F).n).z = +infty and
A4: m <= n;
A5: z in dom(F.m) by A2,A4,Th22;
  consider k be Nat such that
A6: k <= n and
A7: (F.k).z = +infty by A2,A3,Th23;
  z in dom(F.k) by A2,A6,Th22;
  then z in dom(F.m) /\ dom(F.k) by A5,XBOOLE_0:def 4;
  hence thesis by A1,A7;
end;
