reserve r,s,t,u for Real;

theorem Th35:
  for X being LinearTopSpace, a being Point of X holds transl(a,X)
  " = transl(-a,X)
proof
  let X be LinearTopSpace, a be Point of X;
A1: rng transl(a,X) = [#]X by Th34;
  now
    let x be Point of X;
    consider u being object such that
A2: u in dom transl(a,X) and
A3: x = transl(a,X).u by A1,FUNCT_1:def 3;
    reconsider u as Point of X by A2;
A4: x = a+u by A3,Def10;
    transl(a,X) is onto one-to-one by A1,Lm8,FUNCT_2:def 3;
    hence transl(a,X)".x = (transl(a,X) qua Function)".x by TOPS_2:def 4
      .= u by A3,Lm8,FUNCT_2:26
      .= 0.X+u
      .= (-a+a)+u by RLVECT_1:5
      .= -a+x by A4,RLVECT_1:def 3
      .= transl(-a,X).x by Def10;
  end;
  hence thesis by FUNCT_2:63;
end;
