theorem
  f is_simple_func_in S implies f is A-measurable
proof
  assume
A1: f is_simple_func_in S;
  then Im f is_simple_func_in S by MESFUN7C:43;
  then
A2: Im f is A-measurable by MESFUNC6:50;
  Re f is_simple_func_in S by A1,MESFUN7C:43;
  then Re f is A-measurable by MESFUNC6:50;
  hence f is A-measurable by A2,MESFUN6C:def 1;
end;
