
theorem Th13:
  for F being Field, V being VectSp of F for A being Subset of V
  st A is linearly-independent for a being Element of V st a nin the carrier of
  Lin A holds A\/{a} is linearly-independent
proof
  let F be Field;
  let V be VectSp of F;
  let A be Subset of V such that
A1: A is linearly-independent;
A2: the set of all Sum s where s is Linear_Combination of A = the
  carrier of Lin A by VECTSP_7:def 2;
  let a be Element of V;
  set B = A\/{a};
  assume that
A3: a nin the carrier of Lin A and
A4: B is linearly-dependent;
  consider l being Linear_Combination of B such that
A5: Sum l = 0.V and
A6: Carrier l <> {} by A4,VECTSP_7:def 1;
  a in {a} by TARSKI:def 1;
  then
A7: (l!{a}).a = l.a by RANKNULL:25;
  A c= the carrier of Lin A
  proof
    let x be object;
    assume
A8: x in A;
    then reconsider x as Element of V;
    x in Lin A by A8,VECTSP_7:8;
    hence thesis;
  end;
  then a nin A by A3;
  then B\A={a} by XBOOLE_1:88,ZFMISC_1:50;
  then l = (l!A)+(l!{a}) by RANKNULL:27,XBOOLE_1:7;
  then 0.V = Sum (l!A) + Sum (l!{a}) by A5,VECTSP_6:44
    .= Sum (l!A) + (l.a)*a by A7,VECTSP_6:17;
  then
A9: (l.a)*a = - Sum (l!A) by ALGSTR_0:def 13;
A10: (-(l.a)")*(l!A) is Linear_Combination of A by VECTSP_6:31;
  now
    assume l.a <> 0.F;
    then a = ((l.a)")*(-Sum(l!A)) by A9,VECTSP_1:20
      .= -((l.a)" * Sum(l!A)) by VECTSP_1:22
      .= (-(l.a)")*(Sum(l!A)) by VECTSP_1:21
      .= Sum((-(l.a)")*(l!A)) by VECTSP_6:45;
    hence contradiction by A3,A2,A10;
  end;
  then
A11: a nin Carrier l by VECTSP_6:2;
A12: Carrier l c= B by VECTSP_6:def 4;
  Carrier l c= A
  by A11,A12,ZFMISC_1:136;
  then l is Linear_Combination of A by VECTSP_6:def 4;
  hence contradiction by A1,A5,A6,VECTSP_7:def 1;
end;
