
theorem Th45:
for X1,X2 be non empty set, f be PartFunc of [:X1,X2:],ExtREAL,
 x be Element of X1 holds
  ProjPMap1(max+f,x) = max+(ProjPMap1(f,x))
& ProjPMap1(max-f,x) = max-(ProjPMap1(f,x))
proof
   let X1,X2 be non empty set, f be PartFunc of [:X1,X2:],ExtREAL,
   x be Element of X1;
   dom(ProjPMap1(max+f,x)) = X-section(dom(max+f),x)
 & dom(ProjPMap1(max-f,x)) = X-section(dom(max-f),x) by Def3; then
A1:dom(ProjPMap1(max+f,x)) = X-section(dom f,x)
 & dom(ProjPMap1(max-f,x)) = X-section(dom f,x) by MESFUNC2:def 2,def 3;
   dom(max+(ProjPMap1(f,x))) = dom(ProjPMap1(f,x))
 & dom(max-(ProjPMap1(f,x))) = dom(ProjPMap1(f,x))
     by MESFUNC2:def 2,def 3; then
A2:dom(max+(ProjPMap1(f,x))) = X-section(dom f,x)
 & dom(max-(ProjPMap1(f,x))) = X-section(dom f,x) by Def3;
   for y be Element of X2 st y in dom(ProjPMap1(max+f,x))
    holds (ProjPMap1(max+f,x)).y = (max+(ProjPMap1(f,x))).y
   proof
    let y be Element of X2;
    assume A3: y in dom(ProjPMap1(max+f,x)); then
    y in {y where y is Element of X2: [x,y] in dom f}
       by A1,MEASUR11:def 4; then
A4: ex y1 be Element of X2 st y1 = y & [x,y1] in dom f;

    set z = [x,y];
A5: [x,y] in dom(max+f) by A4,MESFUNC2:def 2; then
A6: (ProjPMap1(max+f,x)).y = (max+f).(x,y) by Def3
      .= max(f.z,0) by A5,MESFUNC2:def 2;

    ProjPMap1(f,x).y = f.(x,y) by A4,Def3;
    hence thesis by A6,A1,A3,A2,MESFUNC2:def 2;
   end;
   hence ProjPMap1(max+f,x) = max+(ProjPMap1(f,x)) by A1,A2,PARTFUN1:5;

   for y be Element of X2 st y in dom(ProjPMap1(max-f,x))
    holds (ProjPMap1(max-f,x)).y = (max-(ProjPMap1(f,x))).y
   proof
    let y be Element of X2;
    assume A8: y in dom(ProjPMap1(max-f,x)); then
    y in {y where y is Element of X2: [x,y] in dom f}
      by A1,MEASUR11:def 4; then
A9: ex y1 be Element of X2 st y1 = y & [x,y1] in dom f;

    set z = [x,y];
A10:[x,y] in dom(max-f) by A9,MESFUNC2:def 3; then
A11:(ProjPMap1(max-f,x)).y = (max-f).(x,y) by Def3
      .= max(-(f.z),0) by A10,MESFUNC2:def 3;

    ProjPMap1(f,x).y = f.(x,y) by A9,Def3;
    hence thesis by A11,A1,A2,A8,MESFUNC2:def 3;
   end;
   hence ProjPMap1(max-f,x) = max-(ProjPMap1(f,x)) by A1,A2,PARTFUN1:5;
end;
