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;
reserve l for Linear_Combination of V;

theorem
  for V,W being finite-dimensional VectSp of F, T being
  linear-transformation of V,W st T is one-to-one holds dim V = rank T
proof
  let V,W be finite-dimensional VectSp of F, T be linear-transformation of V,W;
  assume T is one-to-one;
  then ker T = (0).V by Th15;
  then
A1: nullity(T) = 0 by Th16;
  dim V = rank(T) + nullity(T) by Th44
    .= rank(T) by A1;
  hence thesis;
end;
