theorem Th38:
  F is with_the_same_dom & x in dom(F.0) & (for m be Nat holds F.m
is nonnegative) implies (Partial_Sums F)#x is non-decreasing & (Partial_Sums F)
  #x is convergent
proof
  assume
A1: F is with_the_same_dom;
  assume
A2: x in dom(F.0);
  assume
A3: for m be Nat holds F.m is nonnegative;
  for n,m be Nat st m<=n holds ((Partial_Sums F)#x).m <= ((
  Partial_Sums F)#x).n
  proof
    let n,m be Nat;
    assume m <= n;
    then ((Partial_Sums F).m).x <= ((Partial_Sums F).n).x by A1,A2,A3,Th37;
    then ((Partial_Sums F)#x).m <= ((Partial_Sums F).n).x by MESFUNC5:def 13;
    hence thesis by MESFUNC5:def 13;
  end;
  hence ((Partial_Sums F)#x) is non-decreasing by RINFSUP2:7;
  hence thesis by RINFSUP2:37;
end;
