theorem :: MESFUN7C:49

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX holds
( f is_simple_func_in S iff ex F being Finite_Sep_Sequence of S ex c being FinSequence of COMPLEX st
( dom f = union (rng F) & dom F = dom c & ( for n being Nat st n in dom F holds
for x being set st x in F . n holds
(Re f) . x = (Re c) . n ) & ( for n being Nat st n in dom F holds
for x being set st x in F . n holds
(Im f) . x = (Im c) . n ) ) )