reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;

theorem Th15:
  for F being Ring, V, W being VectSp of F,
      T being linear-transformation of V,W st
  T is one-to-one holds ker T = (0).V
proof
  let F be Ring, V, W be VectSp of F,
      T be linear-transformation of V,W;
  reconsider Z = (0).V as Subspace of ker T by VECTSP_4:39;
  assume
A1: T is one-to-one;
  for v being Element of ker T holds v in Z
  proof
    let v be Element of ker T;
A2: v in ker T;
    assume not v in Z; then
A3: not v = 0.V by VECTSP_4:35;
A4: T.(0.V) = 0.W & dom T = [#]V by Th7,Th9;
    reconsider v as Element of V by VECTSP_4:10;
    T.v = 0.W by A2,Th10;
    hence thesis by A1,A3,A4;
  end;
  hence thesis by VECTSP_4:32;
end;
