
theorem LmEMDetX543:
  for E, L being Z_Module,
  K being Linear_Combination of E,
  H being Linear_Combination of L
  st K = H & the ModuleStr of L = the ModuleStr of E
  holds Sum K = Sum H
  proof
    let E, L be Z_Module,
    K be Linear_Combination of E,
    H be Linear_Combination of L;
    assume AS: K = H & the ModuleStr of L = the ModuleStr of E;
    B3: L is Submodule of E by AS,LmEMDetX53;
    B4: Carrier K c= the carrier of L by AS;
    H = K | the carrier of L by AS;
    hence Sum K = Sum H by ZMODUL03:11,B3,B4;
  end;
