
theorem
  for V being RealLinearSpace, W being strict Subspace of V, M being
  Subset of V, v being VECTOR of V st Up(W) = M holds v + W = v + M
proof
  let V be RealLinearSpace;
  let W be strict Subspace of V;
  let M be Subset of V;
  let v be VECTOR of V;
  assume
A1: Up(W) = M;
  for x being object st x in v + M holds x in v + W
  proof
    let x be object;
    assume x in v + M;
    then consider u being Element of V such that
A2: x = v + u and
A3: u in M;
    u in W by A1,A3;
    then x in {v + u9 where u9 is VECTOR of V : u9 in W} by A2;
    hence thesis by RLSUB_1:def 5;
  end;
  then
A4: v + M c= v + W;
  for x being object st x in v + W holds x in v + M
  proof
    let x be object;
    assume x in v + W;
    then x in {v + u where u is VECTOR of V : u in W} by RLSUB_1:def 5;
    then consider u being VECTOR of V such that
A5: x = v + u and
A6: u in W;
    u in M by A1,A6;
    hence thesis by A5;
  end;
  then v + W c= v + M;
  hence thesis by A4;
end;
