reserve k for Element of NAT;
reserve r,r1 for Real;
reserve i for Integer;
reserve q for Rational;
reserve X for set;
reserve f for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for sequence of S;
reserve A for set;
reserve a for ExtReal;
reserve r,s for Real;
reserve n,m for Element of NAT;
reserve X for non empty set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve A,B for Element of S;

theorem
  for X,S,f,A st f is A-measurable holds A /\ eq_dom(f,-infty) in S
proof
  let X,S,f,A;
  assume
A1: f is A-measurable;
  defpred P[Nat,set] means A /\ less_dom(f,(-$1)) = $2;
A2: for n ex y being Element of S st P[n,y]
  proof
    let n;
    reconsider y=A /\ less_dom(f,(-n)) as Element of S by A1;
    take y;
    thus thesis;
  end;
  consider F being sequence of S such that
A3: for n holds P[n,F.n] from FUNCT_2:sch 3(A2);
 for n being Nat holds P[n,F.n]
  proof let n be Nat;
    n in NAT by ORDINAL1:def 12;
   hence thesis by A3;
  end;
  then A /\ eq_dom(f,-infty) = meet rng F by Th25;
 hence thesis;
end;
