reserve r,s,t,u for Real;

theorem
  for X being RealLinearSpace, M being Subset of X, r being non zero
  Real st 0.X in r*M holds 0.X in M
proof
  let X be RealLinearSpace, M be Subset of X, r be non zero Real;
  assume 0.X in r*M;
  then
A1: ex v being Point of X st r*v = 0.X & v in M;
  r * 0.X = 0.X;
  hence thesis by A1,RLVECT_1:36;
end;
