reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th18:
  for I being Basis of V for A being non empty Subset of V st A
  misses I for B being Subset of V st B = I \/ A holds B is linearly-dependent
proof
  let I be Basis of V;
  let A be non empty Subset of V such that
A1: A misses I;
  consider v being object such that
A2: v in A by XBOOLE_0:def 1;
  let B be Subset of V such that
A3: B = I \/ A;
A4: A c= B by A3,XBOOLE_1:7;
  reconsider v as VECTOR of V by A2;
  reconsider Bv = B \ {v} as Subset of V;
A5: I \ {v} c= B \ {v} by A3,XBOOLE_1:7,33;
  not v in I by A1,A2,XBOOLE_0:3;
  then I c= Bv by A5,ZFMISC_1:57;
  then
A6: Lin(I) is Subspace of Lin(Bv) by RLVECT_3:20;
  assume
A7: B is linearly-independent;
  v in Lin(I) by Th13;
  hence contradiction by A7,A2,A4,A6,Th17,RLSUB_1:8;
end;
