theorem Th40: :: MESFUN12:40

for X1, X2 being non empty set
for x being Element of X1
for y being Element of X2
for f being PartFunc of [:X1,X2:],ExtREAL
for er being ExtReal holds
( ( [x,y] in dom f & f . (x,y) = er implies ( y in dom (ProjPMap1 (f,x)) & (ProjPMap1 (f,x)) . y = er ) ) & ( y in dom (ProjPMap1 (f,x)) & (ProjPMap1 (f,x)) . y = er implies ( [x,y] in dom f & f . (x,y) = er ) ) & ( [x,y] in dom f & f . (x,y) = er implies ( x in dom (ProjPMap2 (f,y)) & (ProjPMap2 (f,y)) . x = er ) ) & ( x in dom (ProjPMap2 (f,y)) & (ProjPMap2 (f,y)) . x = er implies ( [x,y] in dom f & f . (x,y) = er ) ) )