theorem :: TMAP_1:5

for A, B being non empty set
for A1, A2 being non empty Subset of A
for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
( ( A1 is Subset of A2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies A1 is Subset of A2 ) & ( A2 is Subset of A1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies A2 is Subset of A1 ) )