reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;

theorem
  (X/\Y) (-) B = (X (-) B)/\(Y (-) B)
proof
  thus (X/\Y) (-) B c= (X (-) B)/\(Y (-) B)
  proof
    let x be object;
    assume x in (X/\Y) (-) B;
    then consider y being Point of T such that
A1: x=y and
A2: B+y c= X/\Y;
    B+y c= Y by A2,XBOOLE_1:18;
    then
A3: x in {y1 where y1 is Point of T:B+y1 c= Y} by A1;
    B+y c= X by A2,XBOOLE_1:18;
    then x in {y1 where y1 is Point of T:B+y1 c= X} by A1;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (X (-) B)/\(Y (-) B);
  then x in X (-) B by XBOOLE_0:def 4;
  then consider y being Point of T such that
A5: x=y and
A6: B+y c= X;
  x in Y (-) B by A4,XBOOLE_0:def 4;
  then
A7: ex y2 being Point of T st x=y2 & B+y2 c= Y;
  B+y c= X/\Y
  by A5,A6,A7,XBOOLE_0:def 4;
  hence thesis by A5;
end;
