
theorem Th46:
for X1,X2 be non empty set, f be PartFunc of [:X1,X2:],ExtREAL,
 y be Element of X2 holds
  ProjPMap2(max+f,y) = max+(ProjPMap2(f,y))
& ProjPMap2(max-f,y) = max-(ProjPMap2(f,y))
proof
   let X1,X2 be non empty set, f be PartFunc of [:X1,X2:],ExtREAL,
   y be Element of X2;
   dom(ProjPMap2(max+f,y)) = Y-section(dom(max+f),y)
 & dom(ProjPMap2(max-f,y)) = Y-section(dom(max-f),y) by Def4; then
A1:dom(ProjPMap2(max+f,y)) = Y-section(dom f,y)
 & dom(ProjPMap2(max-f,y)) = Y-section(dom f,y) by MESFUNC2:def 2,def 3;
   dom(max+(ProjPMap2(f,y))) = dom(ProjPMap2(f,y))
 & dom(max-(ProjPMap2(f,y))) = dom(ProjPMap2(f,y))
     by MESFUNC2:def 2,def 3; then
A2:dom(max+(ProjPMap2(f,y))) = Y-section(dom f,y)
 & dom(max-(ProjPMap2(f,y))) = Y-section(dom f,y) by Def4;
   for x be Element of X1 st x in dom(ProjPMap2(max+f,y))
    holds (ProjPMap2(max+f,y)).x = (max+(ProjPMap2(f,y))).x
   proof
    let x be Element of X1;
    assume A3: x in dom(ProjPMap2(max+f,y)); then
    x in {x where x is Element of X1: [x,y] in dom f}
       by A1,MEASUR11:def 5; then
A4: ex x1 be Element of X1 st x1 = x & [x1,y] in dom f;
    set z = [x,y];
A5: [x,y] in dom(max+f) by A4,MESFUNC2:def 2; then
A6: (ProjPMap2(max+f,y)).x = (max+f).(x,y) by Def4
      .= max(f.z,0) by A5,MESFUNC2:def 2;
    ProjPMap2(f,y).x = f.(x,y) by A4,Def4;
    hence thesis by A6,A1,A3,A2,MESFUNC2:def 2;
   end;
   hence ProjPMap2(max+f,y) = max+(ProjPMap2(f,y)) by A1,A2,PARTFUN1:5;

   for x be Element of X1 st x in dom(ProjPMap2(max-f,y))
    holds (ProjPMap2(max-f,y)).x = (max-(ProjPMap2(f,y))).x
   proof
    let x be Element of X1;
    assume A8: x in dom(ProjPMap2(max-f,y)); then
    x in {x where x is Element of X1: [x,y] in dom f}
      by A1,MEASUR11:def 5; then
A9: ex x1 be Element of X1 st x1 = x & [x1,y] in dom f;
    set z = [x,y];
A10:[x,y] in dom(max-f) by A9,MESFUNC2:def 3; then
A11:(ProjPMap2(max-f,y)).x = (max-f).(x,y) by Def4
      .= max(-(f.z),0) by A10,MESFUNC2:def 3;
    ProjPMap2(f,y).x = f.(x,y) by A9,Def4;
    hence thesis by A11,A1,A8,A2,MESFUNC2:def 3;
   end;
   hence ProjPMap2(max-f,y) = max-(ProjPMap2(f,y)) by A1,A2,PARTFUN1:5;
end;
