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 Th17:
  X (-) (Y+p) = (X (-) Y)+(-p)
proof
  thus X (-) (Y+p) c= (X (-) Y)+(-p)
  proof
    let x be object;
    assume x in X (-) (Y+p);
    then consider y being Point of T such that
A1: x = y and
A2: (Y+p)+y c= X;
    Y+(y+p) c= X by A2,Th16;
    then y+p in {y1 where y1 is Point of T:Y+y1 c= X};
    then y+p-p in (X (-) Y)+(-p);
    hence thesis by A1,Lm2;
  end;
  let x be object;
  assume x in (X (-) Y)+(-p);
  then consider y being Point of T such that
A3: x = y+(-p) and
A4: y in X (-) Y;
  reconsider x as Point of T by A3;
  x+p = y-p+p by A3; then
A5: x+p =y by Lm2;
  ex y2 being Point of T st y = y2 & Y+y2 c= X by A4;
  then Y+p+x c= X by A5,Th16;
  hence thesis;
end;
