reserve x for set,
  K for Ring,
  r for Scalar of K,
  V, M, M1, M2, N for LeftMod of K,
  a for Vector of V,
  m, m1, m2 for Vector of M,
  n, n1, n2 for Vector of N,
  A for Subset of V,
  l for Linear_Combination of A,
  W, W1, W2, W3 for Subspace of V;

theorem
  for v being Vector of the ModuleStr of V st a=v holds r*a = r*v
proof
  let v be Vector of (the ModuleStr of V) such that
A1: a=v;
  thus r*a = (the lmult of V).(r,a) by VECTSP_1:def 12
    .= r*v by A1,VECTSP_1:def 12;
end;
