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 Th45:
  X c= Y implies X (O) B c= Y (O) B & X (o) B c= Y (o) B
proof
  assume
A1: X c= Y;
  thus X (O) B c= Y (O) B
  proof
    let x be object;
    assume x in X (O) B;
    then consider x2,b2 being Point of T such that
A2: x=x2+b2 and
A3: x2 in X (-) B and
A4: b2 in B;
    consider y being Point of T such that
A5: x2=y and
A6: B+y c= X by A3;
    B+y c= Y by A1,A6;
    then x2 in {y1 where y1 is Point of T:B+y1 c= Y}by A5;
    hence thesis by A2,A4;
  end;
  let x be object;
  assume x in X (o) B;
  then consider x2 being Point of T such that
A7: x=x2 and
A8: B+x2 c= X (+) B;
  X (+) B c= Y (+) B by A1,Th9;
  then B+x2 c= Y (+) B by A8;
  hence thesis by A7;
end;
