let A, B be non empty set ; for A1, A2 being non empty Subset of A st A1 misses A2 holds
for f1 being Function of A1,B
for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
let A1, A2 be non empty Subset of A; ( A1 misses A2 implies for f1 being Function of A1,B
for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 ) )
assume A1:
A1 misses A2
; for f1 being Function of A1,B
for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
let f1 be Function of A1,B; for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
let f2 be Function of A2,B; ( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
A1 /\ A2 c= A2
by XBOOLE_1:17;
then reconsider g2 = f2 | (A1 /\ A2) as Function of {},B by A1, FUNCT_2:32;
A1 /\ A2 c= A1
by XBOOLE_1:17;
then reconsider g1 = f1 | (A1 /\ A2) as Function of {},B by A1, FUNCT_2:32;
g1 = g2
;
hence
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
by Def1; verum