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 Th28:
  (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 C by A2,XBOOLE_0:def 4;
    then
A3: x in {y1+y where y1 is Point of T:y1 in C} by A1;
    y2 in B by A2,XBOOLE_0:def 4;
    then x in {y1+y where y1 is Point of T:y1 in B} by A1;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (B+y) /\ (C+y);
  then x in B+y by XBOOLE_0:def 4;
  then consider y3 being Point of T such that
A5: x=y3+y and
A6: y3 in B;
  x in C+y by A4,XBOOLE_0:def 4;
  then consider y2 being Point of T such that
A7: x=y2+y and
A8: y2 in C;
  y2+y-y=y3 by A5,A7,Lm2; then
A9: y2=y3 by Lm2;
  then y2 in B/\C by A6,A8,XBOOLE_0:def 4;
  hence thesis by A5,A9;
end;
