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 Th27:
  (B\/C)+y = (B+y) \/ (C+y)
proof
  thus (B\/C)+y c= (B+y) \/ (C+y)
  proof
    let x be object;
    assume x in (B\/C)+y;
    then consider y2 being Point of T such that
A1: x=y2+y and
A2: y2 in B\/C;
    y2 in B or y2 in C by A2,XBOOLE_0:def 3;
    then x in {y1+y where y1 is Point of T : y1 in B} or x in {y1+y
    where y1 is Point of T:y1 in C} by A1;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in (B+y) \/ (C+y);
  then x in B+y or x in C+y by XBOOLE_0:def 3;
  then consider y2 being Point of T such that
A3: x=y2+y & y2 in B or x=y2+y & y2 in C;
  y2 in B\/C by A3,XBOOLE_0:def 3;
  hence thesis by A3;
end;
