
theorem
  for V being vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty RLSStruct, M being Subset
  of V, r being Real st M is convex holds r*M is convex
proof
  let V be vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty RLSStruct;
  let M be Subset of V;
  let r be Real;
  assume
A1: M is convex;
  for u,v being VECTOR of V, p being Real
    st 0 < p & p < 1 & u in r*M & v
  in r*M holds p*u + (1-p)*v in r*M
  proof
    let u,v be VECTOR of V;
    let p be Real;
    assume that
A2: 0 < p & p < 1 and
A3: u in r*M and
A4: v in r*M;
    consider v9 be Element of V such that
A5: v = r*v9 and
A6: v9 in M by A4;
    consider u9 be Element of V such that
A7: u = r * u9 and
A8: u9 in M by A3;
A9: p*u + (1-p)*v = r*p*u9 + (1-p)*(r*v9) by A7,A5,RLVECT_1:def 7
      .= r*p*u9 + r*(1-p)*v9 by RLVECT_1:def 7
      .= r*(p*u9) + r*(1-p)*v9 by RLVECT_1:def 7
      .= r*(p*u9) + r*((1-p)*v9) by RLVECT_1:def 7
      .= r*(p*u9 + (1-p)*v9) by RLVECT_1:def 5;
    p*u9 + (1-p)*v9 in M by A1,A2,A8,A6;
    hence thesis by A9;
  end;
  hence thesis;
end;
