
theorem Th40:
for X1,X2 be non empty set, x be Element of X1, y be Element of X2,
 f be PartFunc of [:X1,X2:],ExtREAL, er be ExtReal holds
  ( [x,y] in dom f & f.(x,y) = er
       iff y in dom(ProjPMap1(f,x)) & (ProjPMap1(f,x)).y = er ) &
  ( [x,y] in dom f & f.(x,y) = er
       iff x in dom(ProjPMap2(f,y)) & (ProjPMap2(f,y)).x = er )
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, a be ExtReal;
   hereby assume that
A2: [x,y] in dom f and
A3: f.(x,y) = a;
    y in X-section(dom f,x) by A2,Th25;
    hence y in dom(ProjPMap1(f,x)) by Def3;
    hence ProjPMap1(f,x).y = a by A3,Th26;
   end;

   hereby assume that
A4: y in dom(ProjPMap1(f,x)) and
A5: ProjPMap1(f,x).y = a;
   y in X-section(dom f,x) by A4,Def3;
   hence [x,y] in dom f by Th25;
   thus f.(x,y) = a by A4,A5,Th26;
   end;

   hereby assume that
A6: [x,y] in dom f and
A7: f.(x,y) = a;
    x in Y-section(dom f,y) by A6,Th25;
    hence x in dom(ProjPMap2(f,y)) by Def4;
    hence ProjPMap2(f,y).x = a by A7,Th26;
   end;

   assume that
A8: x in dom(ProjPMap2(f,y)) and
A9: ProjPMap2(f,y).x = a;
   x in Y-section(dom f,y) by A8,Def4;
   hence [x,y] in dom f by Th25;
   thus f.(x,y) = a by A8,A9,Th26;
end;
