theorem Th76: :: TOPS_5:76

for I1, I2 being non empty set
for J1 being non-Empty TopSpace-yielding ManySortedSet of I1
for J2 being non-Empty TopSpace-yielding ManySortedSet of I2
for p being Function of I1,I2
for H being ProductHomeo of J1,J2,p
for F being ManySortedFunction of I1 st p is bijective & ( for i being Element of I1 ex f being Function of (J1 . i),((J2 * p) . i) st
( F . i = f & f is being_homeomorphism ) ) & ( for g being Element of (product J1)
for i being Element of I1 holds (H . g) . (p . i) = (F . i) . (g . i) ) holds
for i being Element of I1
for U being Subset of (J1 . i) holds H .: (product ((Carrier J1) +* (i,U))) = product ((Carrier J2) +* ((p . i),((F . i) .: U)))