
theorem
  for V being Abelian non empty RLSStruct, M being Subset of V st M is
  convex
  for r being Real st 0 < r & r < 1 holds (1-r)*M + r*M c= M
proof
  let V be Abelian non empty RLSStruct;
  let M be Subset of V;
  assume
A1: M is convex;
  let r be Real;
  assume
A2: 0 < r & r < 1;
  for x being Element of V st x in (1-r)*M + r*M holds x in M
  proof
    let x be Element of V;
    assume x in (1-r)*M + r*M;
    then x in {u + v where u,v is Element of V : u in (1-r)*M & v in r*M} by
RUSUB_4:def 9;
    then consider u,v be Element of V such that
A3: x = u + v and
A4: u in (1-r)*M & v in r*M;
    (ex w1 be Element of V st u = (1-r) * w1 & w1 in M )& ex w2 be
    Element of V st v = r*w2 & w2 in M by A4;
    hence thesis by A1,A2,A3;
  end;
  hence thesis;
end;
