reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL ,
  A,B for Element of S,
  r,s for Real;

theorem Th80:
  dom f = A implies (f is B-measurable iff f is (A/\B)-measurable)
proof
  assume
A1: dom f = A;
A2: now
    let r be Real;
    now
      let x be object;
      x in A /\ less_dom(f,r) iff x in A & x in less_dom(f,r) by XBOOLE_0:def 4
;
      hence x in A /\ less_dom(f,r) iff x in less_dom(f,r) by A1,
MESFUNC1:def 11;
    end;
    then
    A /\ less_dom(f,r) c= less_dom(f,r) & less_dom(f,r) c= A /\ less_dom(f
    ,r);
    hence A /\ less_dom(f,r) = less_dom(f,r);
  end;
  hereby
    assume
A3: f is B-measurable;
    now
      let r be Real;
      A /\ B /\ less_dom(f,r) = B /\ (A /\ less_dom(f,r)) by XBOOLE_1:16
        .= B /\ less_dom(f,r) by A2;
      hence A/\B/\less_dom(f,r) in S by A3,Th12;
    end;
    hence f is (A/\B)-measurable by Th12;
  end;
  assume
A4: f is (A/\B)-measurable;
  now
    let r be Real;
    A /\ B /\ less_dom(f,r) = B /\ (A /\ less_dom(f,r)) by XBOOLE_1:16
      .=B /\ less_dom(f,r) by A2;
    hence B /\ less_dom(f,r) in S by A4,Th12;
  end;
  hence thesis by Th12;
end;
