
theorem Th19:
for X be non empty set, S be SigmaField of X, f be PartFunc of X,ExtREAL
 st f is empty holds f is_simple_func_in S
proof
   let X be non empty set, S be SigmaField of X, f be PartFunc of X,ExtREAL;
   reconsider EMP = {} as Element of S by MEASURE1:7;
   reconsider F = <*EMP*> as Finite_Sep_Sequence of S;
   assume A1: f is empty; then
   dom f = {} & rng F = {EMP} by FINSEQ_1:38; then
A2:dom f = union rng F by ZFMISC_1:25;
   for n be Nat, x,y be Element of X st n in dom F & x in F.n & y in F.n holds
    f.x = f.y
   proof
    let n be Nat, x,y be Element of X;
    assume A3: n in dom F & x in F.n & y in F.n; then
    n in {1} by FINSEQ_1:2,38; then
    n = 1 by TARSKI:def 1;
    hence f.x = f.y by A3;
   end;
   hence f is_simple_func_in S by A1,A2,MESFUNC2:def 4;
end;
