reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;
reserve f for PartFunc of X,REAL,
  a for Real;

theorem
  for f be PartFunc of X,REAL, a be Real st A c= dom f holds A /\
  great_dom(f,a) = A\(A /\ less_eq_dom(f,a))
proof
  let f be PartFunc of X,REAL, a be Real;
  now
    let x be object;
    assume
A1: x in A /\ great_dom(f,a);
    then x in great_dom(f,a) by XBOOLE_0:def 4;
    then ex y be Real st y = f.x & a < y by MESFUNC6:5;
    then not (ex y1 be Real st y1 = f.x & y1 <= a);
    then not x in less_eq_dom(f,a) by MESFUNC6:4;
    then
A2: not x in (A /\ less_eq_dom(f,a)) by XBOOLE_0:def 4;
    x in A by A1,XBOOLE_0:def 4;
    hence x in A\(A /\ less_eq_dom(f,a)) by A2,XBOOLE_0:def 5;
  end;
  then
A3: A /\ great_dom(f,a) c= A\(A /\ less_eq_dom(f,a)) by TARSKI:def 3;
  assume
A4: A c= dom f;
  now
    let x be object;
    assume
A5: x in A\(A /\ less_eq_dom(f,a));
    then
A6: x in A by XBOOLE_0:def 5;
    not x in A /\ less_eq_dom(f,a) by A5,XBOOLE_0:def 5;
    then not x in less_eq_dom(f,a) by A6,XBOOLE_0:def 4;
    then not f.x <= a by A4,A6,MESFUNC6:4;
    then x in great_dom(f,a) by A4,A6,MESFUNC6:5;
    hence x in A /\ great_dom(f,a) by A6,XBOOLE_0:def 4;
  end;
  then A\(A /\ less_eq_dom(f,a)) c= A /\ great_dom(f,a) by TARSKI:def 3;
  hence thesis by A3,XBOOLE_0:def 10;
end;
