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 & A c= dom f holds
  A /\ great_dom(f,-infty) /\ less_dom(f,+infty) in S
proof
  let X,S,f,A;
  assume that
A1: f is A-measurable and
A2: A c= dom f;
A3: A /\ great_dom(f,-infty) in S
  proof
    defpred P[Element of NAT,set] means A /\ great_dom(f,(-$1)) = $2;
A4: for n ex y being Element of S st P[n,y]
    proof
      let n;
      reconsider y=A /\ great_dom(f,(-n)) as Element of S by A1,A2,Th29;
      take y;
      thus thesis;
    end;
    consider F being sequence of S such that
A5: for n holds P[n,F.n] from FUNCT_2:sch 3(A4);
 A /\ great_dom(f,-infty) = union rng F by A5,Th26;
    hence thesis;
  end;
A6: A /\ less_dom(f,+infty) in S
  proof
    defpred P[Element of NAT,set] means A /\ less_dom(f,$1) = $2;
A7: 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
A8: for n holds P[n,F.n] from FUNCT_2:sch 3(A7);
 A /\ less_dom(f,+infty) = union rng F by A8,Th24;
    hence thesis;
  end;
 (A /\ great_dom(f,-infty)) /\ (A /\ less_dom(f,+infty))
  = (A /\ great_dom(f,-infty) /\ A) /\ less_dom(f,+infty) by XBOOLE_1:16
    .= (great_dom(f,-infty) /\ (A /\ A)) /\ less_dom(f,+infty) by XBOOLE_1:16;
  hence thesis by A3,A6,FINSUB_1:def 2;
end;
