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 A c= dom f holds f is A-measurable iff
  for r being Real holds A /\ great_eq_dom(f,r) in S
proof
  let X,S,f,A;
  assume
A1: A c= dom f;
A2: f is A-measurable implies for r being Real holds A /\
  great_eq_dom(f,r) in S
  proof
    assume
A3: f is A-measurable;
 for r being Real holds A /\ great_eq_dom(f,r) in S
    proof
      let r be Real;
         A /\ less_dom(f,r) in S & A /\ great_eq_dom(f,r) = A\(A /\
      less_dom(f,r)) by A1,A3,Th14;
      hence thesis by MEASURE1:6;
    end;
    hence thesis;
  end;
 (for r being Real holds A /\ great_eq_dom(f,r) in S) implies
  f is A-measurable
  proof
    assume
A4: for r being Real holds A /\ great_eq_dom(f,r) in S;
 for r being Real holds A /\ less_dom(f,r) in S
    proof
      let r be Real;
         A /\ great_eq_dom(f,r) in S & A /\ less_dom(f,r) = A\(A /\
      great_eq_dom(f,r)) by A1,A4,Th17;
      hence thesis by MEASURE1:6;
    end;
    hence thesis;
  end;
  hence thesis by A2;
end;
