
theorem Th32:
for X1,X2 be non empty set, x be Element of X1, y be Element of X2,
  f be PartFunc of [:X1,X2:],ExtREAL
 st f is nonnegative holds
  ProjPMap1(f,x) is nonnegative & ProjPMap2(f,y) is nonnegative
proof
   let X1,X2 be non empty set, x be Element of X1, y be Element of X2,
   f be PartFunc of [:X1,X2:],ExtREAL;
   assume A1: f is nonnegative;

   for q be object st q in dom (ProjPMap1(f,x)) holds 0 <= (ProjPMap1(f,x)).q
   proof
    let q be object;
    assume A2: q in dom(ProjPMap1(f,x)); then
    reconsider y1 = q as Element of X2;
    ProjPMap1(f,x).q = f.(x,y1) by A2,Th26;
    hence 0 <= (ProjPMap1(f,x)).q by A1,SUPINF_2:51;
   end;
   hence ProjPMap1(f,x) is nonnegative by SUPINF_2:52;

   for p be object st p in dom(ProjPMap2(f,y)) holds 0 <= (ProjPMap2(f,y)).p
   proof
    let p be object;
    assume A3: p in dom(ProjPMap2(f,y)); then
    reconsider x1 = p as Element of X1;
    ProjPMap2(f,y).p = f.(x1,y) by A3,Th26;
    hence 0 <= (ProjPMap2(f,y)).p by A1,SUPINF_2:51;
   end;
   hence ProjPMap2(f,y) is nonnegative by SUPINF_2:52;
end;
