
theorem LMBASE2X:
  for K being Ring, V being LeftMod of K, A being finite Subset of V
  holds
  A is linearly-independent
  iff
  ( for L being Linear_Combination of A
  st ( ex F being FinSequence of the carrier of V
  st F is one-to-one & rng F = A
  & Sum(L (#) F) = 0.V)
  holds Carrier(L) = {})
  proof
    let K be Ring, V be LeftMod of K, A be finite Subset of V;
    hereby
      assume
      BS: A is linearly-independent;
      let L be Linear_Combination of A;
      given F be FinSequence of the carrier of V such that
      BS2: F is one-to-one & rng F = A & Sum(L (#) F) = 0.V;
      reconsider B = Carrier L as finite Subset of V;
      set F0 = canFS (B);
      BS3: rng F0 = B by FUNCT_2:def 3;
      rng F0 c= the carrier of V by TARSKI:def 3;
      then reconsider F0 as FinSequence of the carrier of V by FINSEQ_1:def 4;
      reconsider C = A \ B as finite Subset of V;
      set G0 = canFS (C);
      BS4: rng G0 = C by FUNCT_2:def 3;
      rng G0 c= the carrier of V by TARSKI:def 3;
      then reconsider G0 as FinSequence of the carrier of V by FINSEQ_1:def 4;
      BS5: rng F0 /\ rng G0 = B /\ ( A \ B) by BS3,FUNCT_2:def 3
      .= (B /\ A) \ B by XBOOLE_1:49
      .= {} by XBOOLE_1:37,XBOOLE_1:17; then
      BS6: F0^G0 is one-to-one by LMBASE2A;
      BS7: rng (F0^G0) = B \/ (A \ B) by BS3,BS4,BS5,LMBASE2A
      .= A \/ B by XBOOLE_1:39
      .= A by VECTSP_6:def 4,XBOOLE_1:12;
      (rng G0) /\ Carrier(L) = {} by BS5,FUNCT_2:def 3; then
      BS10: L (#) G0 = dom G0 --> 0.V by LMBASE2C; then
aa:   dom (L (#) G0) = dom G0 by FUNCOP_1:13;
      BS12: Sum(L (#) F) = Sum(L (#) (F0^G0)) by EQSUMLF,BS6,BS7,BS2
      .= Sum((L (#) F0)^ (L(#)G0)) by VECTSP_6:13
      .= Sum(L (#) F0) + Sum(L (#) G0) by RLVECT_1:41
      .= Sum(L (#) F0) + 0.V by BS10,LMBASE2D,aa
      .= Sum(L (#) F0);
      Sum(L) = 0.V by BS2,BS3,BS12,VECTSP_6:def 6;
      hence Carrier(L) = {} by VECTSP_7:def 1,BS;
    end;
    assume AS:
    for L being Linear_Combination of A
    st ( ex F being FinSequence of the carrier of V
    st F is one-to-one & rng F = A & Sum(L (#) F) = 0.V)
    holds Carrier(L) = {};
    for L being Linear_Combination of A
    st Sum(L) = 0.V
    holds Carrier(L) = {}
    proof
      let L be Linear_Combination of A;
      assume BS: Sum(L) = 0.V;
      consider F0 be FinSequence of the carrier of V such that
      P3: F0 is one-to-one & rng F0 = Carrier(L)
      & Sum(L) = Sum(L (#) F0) by VECTSP_6:def 6;
      reconsider B = Carrier L as finite Subset of V;
      reconsider C = A \ B as finite Subset of V;
      set G0 = canFS (C);
      BS4: rng G0 = C by FUNCT_2:def 3;
      rng G0 c= the carrier of V by TARSKI:def 3;
      then reconsider G0 as FinSequence of the carrier of V by FINSEQ_1:def 4;
      set F = F0^G0;
      BS5: rng F0 /\ rng G0 = B /\ ( A \ B) by P3,FUNCT_2:def 3
      .= (B /\ A) \ B by XBOOLE_1:49
      .= {} by XBOOLE_1:37,XBOOLE_1:17; then
      BS6: F is one-to-one by LMBASE2A,P3;
      BS7: rng F = B \/ (A \ B) by P3,BS4,BS5,LMBASE2A
      .= A \/ B by XBOOLE_1:39
      .= A by VECTSP_6:def 4,XBOOLE_1:12;
      BS10: L (#) G0 = dom G0 --> 0.V by BS5,P3,LMBASE2C; then
aa:   dom (L (#) G0) = dom G0 by FUNCOP_1:13;
      Sum(L (#) F) = Sum((L (#) F0)^ (L(#)G0)) by VECTSP_6:13
      .= Sum(L (#) F0) + Sum(L (#) G0) by RLVECT_1:41
      .= Sum(L (#) F0) + 0.V by BS10,LMBASE2D,aa
      .= Sum(L (#) F0);
      hence Carrier(L) = {} by AS,BS,BS6,BS7,P3;
    end;
    hence thesis by VECTSP_7:def 1;
  end;
