
theorem Th11:
  for F,G being sequence of ExtREAL st G is nonnegative holds
for S being non empty Subset of NAT holds for H being Function of S,NAT st H is
one-to-one holds (for k being Element of NAT holds ((k in S implies F.k = (G*H)
  .k) & ((not k in S) implies F.k = 0.))) implies SUM(F) <= SUM(G)
proof
  let F,G be sequence of ExtREAL;
  assume
A1: G is nonnegative;
  let S be non empty Subset of NAT;
  let H be Function of S,NAT;
  assume
A2: H is one-to-one;
  assume for k being Element of NAT holds (k in S implies F.k = (G*H).k) & ((
  not k in S) implies F.k = 0.);
  then F = On(G*H) by Def1;
  hence thesis by A1,A2,Th10;
end;
