
theorem LmEMDetX541:
  for E, L being Z_Module, I being Subset of L, J being Subset of E,
  K being Linear_Combination of J
  st I = J & the ModuleStr of L = the ModuleStr of E
  holds K is Linear_Combination of I
  proof
    let E, L be Z_Module, I be Subset of L, J be Subset of E,
    K be Linear_Combination of J;
    assume that AS1: I = J
    and AS2: the ModuleStr of L = the ModuleStr of E;
    P1: K is Linear_Combination of E &
    Carrier K c= J by VECTSP_6:def 4;
    consider T be finite Subset of E such that
    P4: for v being Element of E st not v in T holds
    K.v = 0.(INT.Ring) by VECTSP_6:def 1;
    reconsider S = T as finite Subset of L by AS2;
    reconsider H = K as Linear_Combination of L by AS2,P4,VECTSP_6:def 1;
    Carrier H c= I by P1,AS1,AS2;
    hence thesis by VECTSP_6:def 4;
  end;
