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 Th8:
  for X,Y st Y = {} holds X (-) Y = the carrier of T
proof
  let X,Y;
  assume
A1: Y = {};
  {y where y is Point of T:Y+y c= X} = the carrier of T
  proof
    thus {y where y is Point of T:Y+y c= X} c= the carrier of T
    proof
      let x be object;
      assume x in {y where y is Point of T:Y+y c= X};
      then ex y being Point of T st x=y & Y+y c= X;
      hence thesis;
    end;
    let x be object;
    assume x in the carrier of T;
    then reconsider a=x as Point of T;
    Y+a = {} by A1,Th4;
    then Y+a c= X;
    hence thesis;
  end;
  hence thesis;
end;
