
theorem LmEMDetX51:
  for L, E being Z_Module, I being Subset of L, J being Subset of E
  st the ModuleStr of L = the ModuleStr of E & I = J
  holds (I is linearly-independent iff J is linearly-independent)
  proof
    for E, L being Z_Module, I being Subset of L, J being Subset of E
    st I = J & the ModuleStr of L = the ModuleStr of E
    & I is linearly-independent
    holds J is linearly-independent
    proof
      let E, L be Z_Module, I be Subset of L, J be Subset of E;
      assume that
      A1: I = J and
      A3: the ModuleStr of L = the ModuleStr of E
      and A2: I is linearly-independent;
      for K being Linear_Combination of J st Sum K = 0. E holds
      Carrier K = {}
      proof
        let K be Linear_Combination of J;
        assume P0: Sum K = 0.E;
        reconsider H = K as Linear_Combination of I by A1,A3,LmEMDetX541;
        P1: Carrier K = Carrier H by A3;
        Sum H = 0. L by A3,P0,LmEMDetX543;
        hence thesis by A2,P1,VECTSP_7:def 1;
      end;
      hence thesis by VECTSP_7:def 1;
    end;
    hence thesis;
  end;
