theorem Th15:
  (for m be Nat holds F.m is_simple_func_in S) implies (
  Partial_Sums F).n is_simple_func_in S
proof
  assume
A1: for m be Nat holds F.m is_simple_func_in S;
  for m be Nat holds (R_EAL F).m is_simple_func_in S
  proof
    let m be Nat;
    F.m is_simple_func_in S by A1;
    then R_EAL(F.m) is_simple_func_in S by MESFUNC6:49;
    hence (R_EAL F).m is_simple_func_in S;
  end;
  then (Partial_Sums R_EAL F).n is_simple_func_in S by MESFUNC9:35;
  then (R_EAL(Partial_Sums F)).n is_simple_func_in S by Th7;
  then R_EAL((Partial_Sums F).n) is_simple_func_in S;
  hence (Partial_Sums F).n is_simple_func_in S by MESFUNC6:49;
end;
