reserve r,s,t,u for Real;

theorem Th34:
  for X being LinearTopSpace, a being Point of X holds rng transl( a,X) = [#]X
proof
  let X be LinearTopSpace, a be Point of X;
  thus rng transl(a,X) c= [#]X;
  let y be object;
  assume y in [#]X;
  then reconsider y as Point of X;
  transl(a,X).(-a+y) = a+(-a+y) by Def10
    .= a+-a+y by RLVECT_1:def 3
    .= 0.X+y by RLVECT_1:5
    .= y;
  hence thesis by FUNCT_2:4;
end;
