reserve S for 1-sorted,
  i for Element of NAT,
  p for FinSequence,
  X for set;

theorem Th39:
  for X being finite set holds Lin(singletons(X)) = bspace(X)
proof
  let X be finite set;
  set V = bspace(X);
  set S = singletons(X);
  for v being Element of V holds v in Lin(S)
  proof
    let v be Element of V;
    reconsider f = v as Subset of X;
    consider A being set such that
A1: A c= X and
A2: f = A;
    reconsider A as Subset of X by A1;
    ex l being Linear_Combination of S st Sum l = A by Th38;
    hence thesis by A2,VECTSP_7:7;
  end;
  hence thesis by VECTSP_4:32;
end;
