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,r,s st f is A-measurable & A c= dom f holds
  (A /\ great_dom(f,r) /\ less_dom(f,s)) in S
proof
  let X,S,f,A,r,s;
  assume that
A1: f is A-measurable and
A2: A c= dom f;
A3: A /\ less_dom(f,s) in S by A1;
A4: for r1 being Real holds A /\ great_eq_dom(f,r1) in S
  proof
    let r1 be Real;
     A
 /\ less_dom(f,r1) in S & A /\ great_eq_dom(f,r1) = A\(A /\
    less_dom(f,r1)) by A1,A2,Th14;
    hence thesis by MEASURE1:6;
  end;
 for r1 holds A /\ great_dom(f,r1) in S
  proof
    let r1;
    defpred P[Element of NAT,set] means
    A /\ great_eq_dom(f,(r1+1/($1+1))) = $2;
A5: for n ex y being Element of S st P[n,y]
    proof
      let n;
      reconsider y=A /\ great_eq_dom(f,(r1+1/(n+1))) as Element of S
      by A4;
      take y;
      thus thesis;
    end;
    consider F being sequence of S such that
A6: for n holds P[n,F.n] from FUNCT_2:sch 3(A5);
 A /\ great_dom(f,r1) = union rng F by A6,Th22;
    hence thesis;
  end;
   then A7: A /\ great_dom(f,r) in S;
 (A /\ great_dom(f,r))/\(A /\ less_dom(f,s))
  = (A /\ great_dom(f,r) /\ A) /\ less_dom(f,s) by XBOOLE_1:16
    .= (great_dom(f,r) /\ (A /\ A)) /\
  less_dom(f,s) by XBOOLE_1:16;
  hence thesis by A3,A7,FINSUB_1:def 2;
end;
