reserve r,s,t,u for Real;

theorem
  for X being RealLinearSpace, M being convex Subset of X, r being Real
  st 0 <= r & r <= 1 & 0.X in M holds r*M c= M
proof
  let X be RealLinearSpace, M be convex Subset of X, r be Real such that
A1: 0 <= r & r <= 1 & 0.X in M;
  let x be object;
  assume x in r*M;
  then consider v being Point of X such that
A2: r*v = x and
A3: v in M;
  r*v + (1-r)*0.X in M by A1,A3,Def1;
  then r*v + 0.X in M;
  hence thesis by A2;
end;
