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
  f is B-measurable & A = dom f /\ B implies f|B is A-measurable
proof
  assume that
A1: f is B-measurable and
A2: A = dom f /\ B;
A3: R_EAL f is B-measurable by A1;
  now
    let r be Real;
    now
      let x be object;
      x in dom(f|B) & f|B.x < r iff x in dom f /\ B & f|B.x <
      r by RELAT_1:61;
      then
A4:   x in A & x in less_dom(f|B,r) iff x in B & x in dom f & (f|B).x <
      r by A2,MESFUNC1:def 11,XBOOLE_0:def 4;
      x in B & x in dom f implies (f.x < r iff (f|B).x < r) by
FUNCT_1:49;
      then x in A /\ less_dom(f|B,r) iff x in B & x in less_dom(f,r) by A4,
MESFUNC1:def 11,XBOOLE_0:def 4;
      hence x in A /\ less_dom(f|B,r) iff x in B /\ less_dom(f,r) by
XBOOLE_0:def 4;
    end;
    then A /\ less_dom(f|B,r) c= B /\ less_dom(f,r) & B /\ less_dom(f,r) c= A
    /\ less_dom(f|B,r);
    then A /\ less_dom(f|B,r) = B /\ less_dom(f,r);
    then A /\ less_dom(f|B,r) in S by A3,MESFUNC1:def 16;
    hence A /\ less_dom(f|B,r) in S;
  end;
  hence thesis by Th12;
end;
