
theorem canlinsurj2:
for F being Field
for U,V being finite-dimensional VectSp of F
for B being Basis of U
for f being Function of B,V holds (canLinTrans f) .: B c= rng f
proof
let F be Field, U,V be finite-dimensional VectSp of F;
let B be Basis of U, f be Function of B,V;
set T = canLinTrans f;
now let o be object;
  assume o in (canLinTrans f) .: B; then
  consider x being object such that
  A: x in dom T & x in B & o = T.x by FUNCT_1:def 6;
  B: dom f = B by FUNCT_2:def 1;
  f.x = (T|B).x by defcl .= T.x by A,FUNCT_1:49;
  hence o in rng f by A,B,FUNCT_1:3;
  end;
hence thesis;
end;
