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
  V is finite-dimensional implies for A being Subset of V st A is
  linearly-independent holds A is finite
proof
  assume
A1: V is finite-dimensional;
  let A be Subset of V;
  assume A is linearly-independent;
  then consider B being Basis of V such that
A2: A c= B by Th2;
  B is finite by A1,Th23;
  hence thesis by A2;
end;
