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