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 A-measurable & A c= dom f implies A /\ great_eq_dom(f,r) /\
  less_dom(f,s) in S
proof
  assume that
A1: f is A-measurable and
A2: A c= dom f;
  R_EAL f is A-measurable by A1;
  then
A3: A /\ less_dom(R_EAL f,s) in S by 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,Th13;
  hence thesis by A3,A4,FINSUB_1:def 2;
end;
