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;

theorem
  f is A-measurable & g is A-measurable & A c= dom g implies 
  f-g is A-measurable
proof
  assume that
A1: f is A-measurable and
A2: g is A-measurable and
A3: A c= dom g;
  R_EAL g is A-measurable by A2;
  then (-1)(#)R_EAL g is A-measurable by A3,MESFUNC1:37;
  then -R_EAL g is A-measurable by MESFUNC2:9;
  then
A4: R_EAL -g is A-measurable by Th28;
  R_EAL f is A-measurable by A1;
  then R_EAL f + R_EAL -g is A-measurable by A4,MESFUNC2:7;
  then R_EAL(f-g) is A-measurable by Th23;
  hence thesis;
end;
