
theorem
for X1,X2 be non empty set, f be PartFunc of [:X1,X2:],ExtREAL,
 x be Element of X1, y be Element of X2
 st (for z be Element of [:X1,X2:] st z in dom f holds f.z = 0)
 holds ProjPMap2(f,y).x = 0 & ProjPMap1(f,x).y = 0
proof
   let X1,X2 be non empty set, f be PartFunc of [:X1,X2:],ExtREAL,
   x be Element of X1, y be Element of X2;
   assume
A1: for z be Element of [:X1,X2:] st z in dom f holds f.z = 0;

   now assume x in dom(ProjPMap2(f,y)); then
    x in Y-section(dom f,y) by Def4; then
    x in {x where x is Element of X1: [x,y] in dom f} by MEASUR11:def 5; then
    consider x1 be Element of X1 such that
A2:  x1 = x & [x1,y] in dom f;
    f.(x1,y) = 0 by A1,A2;
    hence ProjPMap2(f,y).x = 0 by A2,Def4;
   end;
   hence ProjPMap2(f,y).x = 0 by FUNCT_1:def 2;

   now assume y in dom(ProjPMap1(f,x)); then
    y in X-section(dom f,x) by Def3; then
    y in {y where y is Element of X2: [x,y] in dom f} by MEASUR11:def 4; then
    consider y1 be Element of X2 such that
A3:  y1 = y & [x,y1] in dom f;
    f.(x,y1) = 0 by A1,A3;
    hence ProjPMap1(f,x).y = 0 by A3,Def3;
   end;
   hence ProjPMap1(f,x).y = 0 by FUNCT_1:def 2;
end;
