
theorem Th25:
  for V being RealUnitarySpace, A being Subset of V st A is
linearly-independent for v being VECTOR of V st v in A for B being Subset of V
  st B = A \ {v} holds not v in Lin(B)
proof
  let V be RealUnitarySpace;
  let A be Subset of V such that
A1: A is linearly-independent;
  let v be VECTOR of V;
  assume v in A;
  then
A2: {v} is Subset of A by SUBSET_1:41;
  let B be Subset of V such that
A3: B = A \ {v};
  B c= A by A3,XBOOLE_1:36;
  then
A4: B \/ {v} c= A \/ A by A2,XBOOLE_1:13;
  assume v in Lin(B);
  then consider L being Linear_Combination of B such that
A5: v = Sum(L) by Th1;
  {v} is linearly-independent by A1,A2,RLVECT_3:5;
  then v <> 0.V by RLVECT_3:8;
  then Carrier(L) is non empty by A5,RLVECT_2:34;
  then
A6: ex w being object st w in Carrier(L);
  v in {v} by TARSKI:def 1;
  then v in Lin({v}) by Th2;
  then consider K being Linear_Combination of {v} such that
A7: v = Sum(K) by Th1;
A8: Carrier(L) c= B & Carrier(K) c= {v} by RLVECT_2:def 6;
  then Carrier(L) \/ Carrier(K) c= B \/ {v} by XBOOLE_1:13;
  then Carrier(L - K) c= Carrier(L) \/ Carrier(K) & Carrier(L) \/ Carrier(K)
  c= A by A4,RLVECT_2:55;
  then Carrier(L - K) c= A;
  then
A9: L - K is Linear_Combination of A by RLVECT_2:def 6;
A10: for x being VECTOR of V holds x in Carrier(L) implies K.x = 0
  proof
    let x be VECTOR of V;
    assume x in Carrier(L);
    then not x in Carrier(K) by A3,A8,XBOOLE_0:def 5;
    hence thesis;
  end;
  now
    let x be object such that
A11: x in Carrier(L) and
A12: not x in Carrier(L - K);
    reconsider x as VECTOR of V by A11;
A13: L.x <> 0 by A11,RLVECT_2:19;
    (L - K).x = L.x - K.x by RLVECT_2:54
      .= L.x - 0 by A10,A11
      .= L.x;
    hence contradiction by A12,A13;
  end;
  then
A14: Carrier(L) c= Carrier(L - K);
  0.V = Sum(L) + (- Sum(K)) by A5,A7,RLVECT_1:5
    .= Sum(L) + Sum(-K) by RLVECT_3:3
    .= Sum(L - K) by RLVECT_3:1;
  hence contradiction by A1,A6,A9,A14;
end;
