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 y1,y2 being Point of T such that
A1: x=y1+y2 and
A2: y1 in X\/Y and
A3: y2 in B;
    y1 in X or y1 in Y by A2,XBOOLE_0:def 3;
    then
    x in {y3+y4 where y3,y4 is Point of T:y3 in X & y4 in B} or x
    in {y3+y4 where y3,y4 is Point of T:y3 in Y & y4 in B} by A1,A3;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in (X (+) B)\/(Y (+) B);
  then x in X (+) B or x in Y (+) B by XBOOLE_0:def 3;
  then consider y1,y2 being Point of T such that
A4: x=y1+y2 & y1 in X & y2 in B or x=y1+y2 & y1 in Y & y2 in B;
  y1 in X\/Y by A4,XBOOLE_0:def 3;
  hence thesis by A4;
end;
