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 Th34:
  (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 Y by A2,XBOOLE_0:def 4;
  then
A4: x in {y3+y4 where y3,y4 is Point of T:y3 in Y & y4 in B} by A1,A3;
  y1 in X by A2,XBOOLE_0:def 4;
  then
  x in {y3+y4 where y3,y4 is Point of T:y3 in X & y4 in B} by A1,A3;
  hence thesis by A4,XBOOLE_0:def 4;
end;
