
theorem Th4:
  for V being non empty RLSStruct, M being Subset of V holds M is
  convex iff for r being Real
   st 0 < r & r < 1 holds r*M + (1-r)*M c= M
proof
  let V be non empty RLSStruct;
  let M be Subset of V;
  thus  M is convex implies for r being Real
    st 0 < r & r < 1 holds r*M + (1-r)*
  M c= M
  proof
    assume
A1: M is convex;
    let r be Real;
    assume
A2: 0 < r & r < 1;
    for x being Element of V st x in r*M + (1-r)*M holds x in M
    proof
      let x be Element of V;
      assume x in r*M + (1-r)*M;
      then
      x in {u + v where u,v is Element of V : u in r*M & v in (1-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 r*M & v in (1-r)*M;
      (ex w1 be Element of V st u = r * w1 & w1 in M )& ex w2 be Element
      of V st v = (1-r)*w2 & w2 in M by A4;
      hence thesis by A1,A2,A3;
    end;
    hence thesis;
  end;
    assume
A5: for r being Real st 0 < r & r < 1 holds r*M + (1-r)*M c= M;
    let u,v be VECTOR of V;
    let r be Real;
    assume 0 < r & r < 1;
    then
A6: r*M + (1-r)*M c= M by A5;
    assume u in M & v in M;
    then r*u in r*M & (1-r)*v in {(1-r)*w where w is Element of V: w in M};
    then
    r*u + (1-r)*v in {p+q where p,q is Element of V: p in r*M & q in (1-r )*M};
    then r*u + (1-r)*v in r*M + (1-r)*M by RUSUB_4:def 9;
    hence thesis by A6;
end;
