reserve X for non empty set;
reserve e for set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for Function of RAT,S;
reserve p,q for Rational;
reserve r for Real;
reserve n,m for Nat;
reserve A,B for Element of S;

theorem
  for f,g,A st f is real-valued & g is real-valued & f is A-measurable &
  g is A-measurable & A c= dom g holds f-g is A-measurable
proof
  let f,g,A;
  assume that
A1: f is real-valued and
A2: g is real-valued and
A3: f is A-measurable and
A4: g is A-measurable & A c= dom g;
A5: (-1)(#)g is real-valued by A2,Th10;
A6: (-1)(#)g is A-measurable by A4,MESFUNC1:37;
A7: -g is real-valued by A5,Th9;
 -g is A-measurable by A6,Th9;
then  f+(-g) is A-measurable by A1,A3,A7,Th7;
  hence thesis by Th8;
end;
