
theorem LmEMDetX53:
  for L, E being Z_Module st the ModuleStr of L = the ModuleStr of E
  holds L is Submodule of E
  proof
    let L, E be Z_Module;
    assume AS: the ModuleStr of L = the ModuleStr of E;
    P2: 0.L = 0.E by AS;
    P3: the addF of L = (the addF of E) || the carrier of L by AS;
    the lmult of L
    = (the lmult of E) | ( [: the carrier of INT.Ring, the carrier of L:]
    qua set) by AS;
    hence thesis by AS,P2,P3,VECTSP_4:def 2;
  end;
