
theorem Th12:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,ExtREAL, F be Finite_Sep_Sequence of S,
 a be FinSequence of ExtREAL, n be Nat
st f is_simple_func_in S & F,a are_Re-presentation_of f & n in dom F
holds F.n = {} or a.n is Real
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,ExtREAL, F be Finite_Sep_Sequence of S,
    a be FinSequence of ExtREAL, n be Nat;
    assume that
A1:  f is_simple_func_in S and
A2:  F,a are_Re-presentation_of f and
A3:  n in dom F;

A4: f is real-valued by A1,MESFUNC2:def 4;

    now assume F.n <> {}; then
     consider x be object such that
A5:   x in F.n by XBOOLE_0:def 1;
     a.n = f.x by A2,A3,A5,MESFUNC3:def 1;
     hence a.n is Real by A4;
    end;
    hence F.n = {} or a.n is Real;
end;
