
theorem Th32:
for X be non empty set, f be nonpositive PartFunc of X,ExtREAL
 holds f = -(max-f)
proof
   let X be non empty set, f be nonpositive PartFunc of X,ExtREAL;
b1:dom f = dom(max-f) by MESFUNC2:def 3 .= dom(-(max-f)) by MESFUNC1:def 7;
b3:dom f = dom(max+f) by MESFUNC2:def 2;
   for x be Element of X st x in dom f holds f.x = (-(max-f)).x
   proof
    let x be Element of X;
    assume b2: x in dom f;
    max(f.x,0) = 0 by MESFUNC5:8,XXREAL_0:def 10; then
    (max+f).x = 0 by b2,b3,MESFUNC2:def 2; then
    (max-f).x = -(f.x) by b2,MESFUNC2:20; then
    f.x = -(max-f).x;
    hence f.x = (-(max-f)).x by b1,b2,MESFUNC1:def 7;
   end;
   hence f = -(max-f) by b1,PARTFUN1:5;
end;
