reserve r,s,t,u for Real;

theorem Th13:
  for X being RealLinearSpace, r being Real holds r*{0.X} = {0.X}
proof
  let X be RealLinearSpace, r be Real;
  thus r*{0.X} c= {0.X}
  proof
    let x be object;
    assume
A1: x in r*{0.X};
    then reconsider x as Point of X;
    consider v being Point of X such that
A2: x = r*v and
A3: v in {0.X} by A1;
    v = 0.X by A3,TARSKI:def 1;
    then r*v = 0.X;
    hence thesis by A2,TARSKI:def 1;
  end;
  0.X in {0.X} by TARSKI:def 1;
  then r*0.X in r*{0.X};
  then 0.X in r*{0.X};
  hence thesis by ZFMISC_1:31;
end;
