
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f,g be PartFunc of X,ExtREAL st f is_simple_func_in S &
g is_simple_func_in S holds f-g is_simple_func_in S
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g
  be PartFunc of X,ExtREAL such that
A1: f is_simple_func_in S and
A4: g is_simple_func_in S;
    (-1)(#)g is_simple_func_in S by A4,MESFUNC5:39; then
    -g is_simple_func_in S by MESFUNC2:9; then
    f+(-g) is_simple_func_in S by A1,Th28; then
    -(g-f) is_simple_func_in S by MEASUR11:64;
    hence f-g is_simple_func_in S by MEASUR11:64;
end;
