theorem
  (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 (Im F).m is_simple_func_in S
  proof
    let m be Nat;
    F.m is_simple_func_in S by A1;
    then Im(F.m) is_simple_func_in S by MESFUN7C:43;
    hence (Im F).m is_simple_func_in S by MESFUN7C:24;
  end;
  then (Partial_Sums Im F).n is_simple_func_in S by Th15;
  then (Im(Partial_Sums F)).n is_simple_func_in S by Th29;
  then
A2: Im((Partial_Sums F).n) is_simple_func_in S by MESFUN7C:24;
  for m be Nat holds (Re F).m is_simple_func_in S
  proof
    let m be Nat;
    F.m is_simple_func_in S by A1;
    then Re(F.m) is_simple_func_in S by MESFUN7C:43;
    hence (Re F).m is_simple_func_in S by MESFUN7C:24;
  end;
  then (Partial_Sums Re F).n is_simple_func_in S by Th15;
  then (Re(Partial_Sums F)).n is_simple_func_in S by Th29;
  then Re((Partial_Sums F).n) is_simple_func_in S by MESFUN7C:24;
  hence (Partial_Sums F).n is_simple_func_in S by A2,MESFUN7C:43;
end;
