theorem :: TMAP_1:4

for A, B being non empty set
for A1, A2, A3, A12, A23 being non empty Subset of A st A12 = A1 \/ A2 & A23 = A2 \/ A3 holds
for f1 being Function of A1,B
for f2 being Function of A2,B
for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23