
theorem Th4:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,ExtREAL st f is_simple_func_in S holds
  max+f is_simple_func_in S & max-f is_simple_func_in S
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,ExtREAL;
    assume A1: f is_simple_func_in S; then
A2: f is real-valued by MESFUNC2:def 4;
    consider F be Finite_Sep_Sequence of S such that
A3:  dom f = union rng F and
A4:  for n be Nat, x,y be Element of X st n in dom F & x in F.n & y in F.n
      holds f.x = f.y by A1,MESFUNC2:def 4;

A5: dom max+f = union rng F by A3,MESFUNC2:def 2;
    for n be Nat, x,y be Element of X st n in dom F & x in F.n & y in F.n
     holds (max+f).x = (max+f).y
    proof
     let n be Nat, x,y be Element of X;
     assume that
A6:   n in dom F and
A7:   x in F.n and
A8:  y in F.n;
     F.n in rng F by A6,FUNCT_1:3; then
     x in dom max+f & y in dom max+f by A5,A7,A8,TARSKI:def 4; then
     (max+f).x = max(f.x,0) & (max+f).y = max(f.y,0) by MESFUNC2:def 2;
     hence (max+f).x = (max+f).y by A4,A6,A7,A8;
    end;
    hence max+f is_simple_func_in S by A2,A5,Th2,MESFUNC2:def 4;
A9: dom max-f = union rng F by A3,MESFUNC2:def 3;
    for n be Nat, x,y be Element of X st n in dom F & x in F.n & y in F.n
     holds (max-f).x = (max-f).y
    proof
     let n be Nat, x,y be Element of X;
     assume that
A10:   n in dom F and
A11:   x in F.n and
A12:  y in F.n;
     F.n in rng F by A10,FUNCT_1:3; then
     x in dom max-f & y in dom max-f by A9,A11,A12,TARSKI:def 4; then
     (max-f).x = max(-(f.x),0) & (max-f).y = max(-(f.y),0) by MESFUNC2:def 3;
     hence (max-f).x = (max-f).y by A4,A10,A11,A12;
    end;
    hence max-f is_simple_func_in S by A2,A9,Th2,MESFUNC2:def 4;
end;
