
theorem Th12:
  for V being vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty RLSStruct, M being
  Subset of V, r1,r2 being Real holds (r1 + r2)*M c= r1*M + r2*M
proof
  let V be vector-distributive scalar-distributive scalar-associative
  scalar-unital
   non empty RLSStruct;
  let M be Subset of V;
  let r1,r2 be Real;
  for x being VECTOR of V st x in (r1+r2)*M holds x in r1*M + r2*M
  proof
    let x be VECTOR of V;
    assume x in (r1+r2)*M;
    then consider w be VECTOR of V such that
A1: x = (r1 + r2)*w and
A2: w in M;
A3: r2*w in {r2*u where u is VECTOR of V : u in M} by A2;
    x = r1*w + r2*w & r1*w in r1*M by A1,A2,RLVECT_1:def 6;
    then x in {u + v where u,v is VECTOR of V : u in r1*M & v in r2*M} by A3;
    hence thesis by RUSUB_4:def 9;
  end;
  hence thesis;
end;
