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
A1: F is additive;
  assume that
A2: z in dom((Partial_Sums F).n) and
A3: ((Partial_Sums F).n).z = -infty;
  assume m <= n;
  then
A4: z in dom(F.m) by A2,Th22;
  consider k be Nat such that
A5: k <= n and
A6: (F.k).z = -infty by A2,A3,Th25;
  z in dom(F.k) by A2,A5,Th22;
  then z in dom(F.m) /\ dom(F.k) by A4,XBOOLE_0:def 4;
  hence thesis by A1,A6;
end;
