
theorem ZMODUL033:
for F being Field,
    V being VectSp of F
for X being Subset of V
holds X is linearly-independent iff
      for l1,l2 being Linear_Combination of X st Sum l1 = Sum l2 holds l1 = l2
  proof
let F be Field, V be VectSp of F; let X be Subset of V;
A: now assume A1: X is linearly-independent;
   now let KL1,KL2 be Linear_Combination of X;
    A2: Carrier(KL1) c= X by VECTSP_6:def 4;
    Carrier(KL2) c= X by VECTSP_6:def 4; then
    A3: Carrier(KL1) \/ Carrier(KL2) c= X by A2,XBOOLE_1:8;
    assume Sum(KL1) = Sum(KL2);
    then Sum(KL1) - Sum(KL2) = 0.V by RLVECT_1:5; then
    A4: KL1 - KL2 is Linear_Combination of Carrier(KL1 - KL2) &
        Sum(KL1 - KL2) = 0.V by VECTSP_6:47,VECTSP_6:def 4;
    Carrier(KL1 - KL2) c= Carrier(KL1) \/ Carrier(KL2) by VECTSP_6:41; then
    A5: Carrier(KL1 - KL2) is linearly-independent
        by A1,A3,XBOOLE_1:1,VECTSP_7:1;
    now
      let v be Vector of V;
      not v in Carrier(KL1 - KL2) by VECTSP_7:def 1,A5,A4;
      then (KL1 - KL2).v = 0.F by VECTSP_6:2;
      then KL1.v - KL2.v = 0.F by VECTSP_6:40;
      hence KL1.v = KL2.v by RLVECT_1:21;
    end;
    hence KL1 = KL2;
    end;
    hence for l1,l2 being Linear_Combination of X
         st Sum l1 = Sum l2 holds l1 = l2;
    end;
now assume A1:
   for l1,l2 being Linear_Combination of X st Sum l1 = Sum l2 holds l1 = l2;
   now let l be Linear_Combination of X;
     assume A2: Sum(l) = 0.V;
     A3: Sum(ZeroLC(V)) = 0.V by VECTSP_6:15;
     Carrier(ZeroLC(V)) c= X by VECTSP_6:def 3; then
     ZeroLC(V) is Linear_Combination of X by VECTSP_6:def 4;
     hence Carrier(l) = {} by A1,A2,A3,VECTSP_6:def 3;
     end;
   hence X is linearly-independent by VECTSP_7:def 1;
   end;
hence thesis by A;
end;
