reserve
  A,B,X for set,
  S for SigmaField of X;
reserve C for C_Measure of X;

theorem Th10:
  A in sigma_Field(C) & B in sigma_Field(C) implies A \ B in sigma_Field(C)
proof
  assume that
A1: A in sigma_Field(C) and
A2: B in sigma_Field(C);
  for x being object holds x in A \ B iff x in A /\ (X \ B)
  proof
    let x be object;
    hereby
      assume
A3:   x in A \ B;
      then
A4:   not x in B by XBOOLE_0:def 5;
      x in A by A3;
      then x in (X \ B) by A1,A4,XBOOLE_0:def 5;
      hence x in A /\ (X \ B) by A3,XBOOLE_0:def 4;
    end;
    assume
A5: x in A /\ (X \ B);
    then x in (X \ B) by XBOOLE_0:def 4;
    then
A6: not x in B by XBOOLE_0:def 5;
    x in A by A5,XBOOLE_0:def 4;
    hence thesis by A6,XBOOLE_0:def 5;
  end;
  then
A7: A \ B = A /\ (X \ B) by TARSKI:2;
  X \ B in sigma_Field(C) by A2,Th7;
  hence thesis by A1,A7,Th9;
end;
