
theorem Th26:
for X1,X2 be non empty set, Y be set, f be PartFunc of [:X1,X2:],Y,
 x be Element of X1, y be Element of X2 holds
  ( x in dom ProjPMap2(f,y) implies ProjPMap2(f,y).x = f.(x,y) )
& ( y in dom ProjPMap1(f,x) implies ProjPMap1(f,x).y = f.(x,y) )
proof
   let X1,X2 be non empty set, Y be set, f be PartFunc of [:X1,X2:],Y,
   c be Element of X1, d be Element of X2;
   hereby assume c in dom ProjPMap2(f,d); then
    c in Y-section(dom f,d) by Def4; then
    c in {x where x is Element of X1: [x,d] in dom f} by MEASUR11:def 5; then
    ex x be Element of X1 st c = x & [x,d] in dom f;
    hence ProjPMap2(f,d).c = f.(c,d) by Def4;
   end;
   assume d in dom ProjPMap1(f,c); then
   d in X-section(dom f,c) by Def3; then
   d in {y where y is Element of X2: [c,y] in dom f} by MEASUR11:def 4; then
   ex y be Element of X2 st d=y & [c,y] in dom f;
   hence ProjPMap1(f,c).d = f.(c,d) by Def3;
end;
