
theorem canlinsurj:
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 im(canLinTrans f) = Lin(rng f)
proof
let F be Field, U be non trivial finite-dimensional VectSp of F;
let V be finite-dimensional VectSp of F, B be Basis of U;
let f be Function of B,V;
set T = canLinTrans f;
B: [#](im T) = T .: [#]U by RANKNULL:def 2;
C: T .: [#]U is Subset of (im T) by RANKNULL:12;
G: the carrier of Lin(rng f) = the set of all Sum l where
                    l is Linear_Combination of (rng f) &
   the carrier of Lin B = the set of all Sum l where
                    l is Linear_Combination of B by VECTSP_7:def 2;
H: Lin B = the ModuleStr of U by VECTSP_7:def 3;
F: now let o be object;
   assume o in the carrier of (im T); then
   consider u being object such that
   F1: u in dom T & u in [#]U & o = T.u by B,FUNCT_1:def 6;
   reconsider u as Element of U by F1;
   u in Lin B by H; then
   consider l being Linear_Combination of B such that
   F2: Sum l = u by G;
   T .: B c= (rng f) by canlinsurj2;
   then o in Lin(rng f) by F1,F2,canlinsurj1;
   hence o in the carrier of Lin(rng f);
   end;
now let o be object;
   assume o in the carrier of Lin(rng f); then
   consider l2 being Linear_Combination of (rng f) such that
   G1: o = Sum l2 by G;
   consider l1 being Linear_Combination of B such that
   G2: T.(Sum l1)  = Sum l2 by canlinsurj3;
   dom T = the carrier of U by FUNCT_2:def 1; then
   Sum l2 in T .: [#]U by G2,FUNCT_1:def 6;
   hence o in the carrier of (im T) by G1,C;
   end;
hence thesis by F,TARSKI:2,VECTSP_4:29;
end;
