theorem
  f is_simple_func_in S & g is_simple_func_in S implies f+g is_simple_func_in S
proof
  assume
A1: f is_simple_func_in S & g is_simple_func_in S;
  then Im f is_simple_func_in S & Im g is_simple_func_in S by MESFUN7C:43;
  then Im f + Im g is_simple_func_in S by MESFUNC6:72;
  then
A2: Im(f+g) is_simple_func_in S by MESFUN6C:5;
  Re f is_simple_func_in S & Re g is_simple_func_in S by A1,MESFUN7C:43;
  then Re f + Re g is_simple_func_in S by MESFUNC6:72;
  then Re(f+g) is_simple_func_in S by MESFUN6C:5;
  hence thesis by A2,MESFUN7C:43;
end;
