theorem Th67:
for F be Functional_Sequence of X,REAL st
 (for m be Nat holds F.m is nonnegative)
holds
 for m be Nat holds (Partial_Sums F).m is nonnegative
proof
   let F be Functional_Sequence of X,REAL;
   assume A1: for m be Nat holds F.m is nonnegative;
   defpred P[Nat] means (Partial_Sums F).$1 is nonnegative;
   (Partial_Sums F).0 = F.0 by MESFUN9C:def 2; then
A2:P[ 0] by A1;
A3:now let k be Nat;
    assume P[k]; then
A4:(Partial_Sums F).k is nonnegative & F.(k+1) is nonnegative by A1;
    (Partial_Sums F).(k+1) = (Partial_Sums F).k + F.(k+1) by MESFUN9C:def 2;
    hence P[k+1] by A4,MESFUNC6:56;
   end;
   for k being Nat holds P[k] from NAT_1:sch 2(A2,A3);
   hence thesis;
end;
