
theorem Th33:
  for X be non empty set, S be SigmaField of X,
      f be PartFunc of X ,ExtREAL, A be Element of S,
      r,s be Real st f is A-measurable & A c= dom f
  holds A /\ great_eq_dom(f, r) /\ less_dom(f, s) in S
proof
  let X be non empty set;
  let S be SigmaField of X;
  let f be PartFunc of X,ExtREAL;
  let A be Element of S;
  let r,s be Real;
  assume that
A1: f is A-measurable and
A2: A c= dom f;
A3: A /\ less_dom(f, s) in S by A1,MESFUNC1:def 16;
A4: A /\ great_eq_dom(f, r) /\ (A /\ less_dom(f, s)) = A /\
  great_eq_dom(f, r) /\ A /\ less_dom(f, s) by XBOOLE_1:16
    .= great_eq_dom(f, r) /\ (A/\A) /\ less_dom(f, s) by XBOOLE_1:16;
  A /\ great_eq_dom(f, r) in S by A1,A2,MESFUNC1:27;
  hence thesis by A3,A4,FINSUB_1:def 2;
end;
