reserve E,X,Y,x for set;
reserve A,B,C for Subset of E;

theorem
  (for x being Element of E holds x in A iff not(x in B iff x in C))
  implies A = B \+\ C
proof
  assume
A1: for x being Element of E holds x in A iff not(x in B iff x in C);
  now
    let x be Element of E;
    assume x in A;
    then x in B & not x in C or x in C & not x in B by A1;
    then x in B \ C or x in C \ B by XBOOLE_0:def 5;
    hence x in B \+\ C by XBOOLE_0:def 3;
  end;
  hence A c= B \+\ C by Th2;
  now
    let x be Element of E;
    assume x in B \+\ C;
    then x in B \ C or x in C \ B by XBOOLE_0:def 3;
    then x in B & not x in C or x in C & not x in B by XBOOLE_0:def 5;
    hence x in A by A1;
  end;
  hence thesis by Th2;
end;
