theorem Th3: :: MESFUNC3:3

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL st f is_simple_func_in S holds
ex F being Finite_Sep_Sequence of S ex a being FinSequence of ExtREAL st
( dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being object st x in F . n holds
f . x = a . n ) & ( for x being object st x in dom f holds
ex ax being FinSequence of ExtREAL st
( dom ax = dom a & ( for n being Nat st n in dom ax holds
ax . n = (a . n) * ((chi ((F . n),X)) . x) ) ) ) )