
theorem canLininj:
for F being Field
for U being non trivial finite-dimensional VectSp of F
for V being finite-dimensional VectSp of F
for B being Basis of U
for f being Function of B,V
holds canLinTrans f is one-to-one iff
      (rng f is linearly-independent & f is one-to-one)
proof
let F be Field, U be non trivial finite-dimensional VectSp of F,
    V be finite-dimensional VectSp of F;
let B be Basis of U, f be Function of B,V;
now assume A: canLinTrans f is one-to-one;
  f = (canLinTrans f)|B by defcl;
  hence rng f is linearly-independent &
        f is one-to-one by A,FUNCT_1:52,canLininjrngA;
  end;
hence thesis by canLininjrngB;
end;
