
theorem CLS:
for F being Field
for U being non trivial finite-dimensional VectSp of F
for V being VectSp of F
for B being Basis of U
for f being Function of B,V
for l being Linear_Combination of B
holds (canLinTrans f).(Sum l) = Sum(f (#) l)
proof
let F be Field, U be non trivial finite-dimensional VectSp of F,
    V be VectSp of F;
let B be Basis of U, f be Function of B,V, l be Linear_Combination of B;
set B1 = B;
B: Lin B = the ModuleStr of U by VECTSP_7:def 3;
defpred P[object,object] means
  for l being Linear_Combination of B1 st $1 = Sum l holds $2 = Sum(f (#) l);
AA: now let x be object;
    assume x in the carrier of U; then
    reconsider u = x as Element of U;
    u in the carrier of (Lin B) by B; then
    u in the set of all Sum(l) where l is Linear_Combination of B
      by VECTSP_7:def 2; then
    consider l being Linear_Combination of B1 such that C1: u = Sum(l);
    thus ex y being object st y in the carrier of V & P[x,y]
      proof
      take Sum(f (#) l);
      thus thesis by C1,FIELD_7:6;
      end;
    end;
consider T be Function of the carrier of U, the carrier of V such that
A: for u being object st u in the carrier of U holds P[u,T.u]
   from FUNCT_2:sch 1(AA);
C: now let u1,u2 be Element of U;
   u1 in the carrier of (Lin B) by B; then
   u1 in the set of all Sum(l) where l is Linear_Combination of B
     by VECTSP_7:def 2; then
   consider l1 being Linear_Combination of B1 such that
   C1: u1 = Sum(l1);
   u2 in the carrier of (Lin B) by B; then
   u2 in the set of all Sum(l) where l is Linear_Combination of B
     by VECTSP_7:def 2; then
   consider l2 being Linear_Combination of B1 such that
   C2: u2 = Sum(l2);
   reconsider l3 = l1 + l2 as Linear_Combination of B1 by VECTSP_6:24;
   thus T.(u1) + T.(u2)
      = Sum(f (#) l1) + T.(Sum l2) by A,C1,C2
     .= Sum(f (#) l1) + Sum(f (#) l2) by A
     .= Sum(f (#) l3) by lemadd
     .= T.(Sum(l1+l2)) by A
     .= T.(u1+u2) by C1,C2,VECTSP_6:44;
  end;
now let  a being Element of F, u be Element of U;
  u in the carrier of (Lin B) by B; then
  u in the set of all Sum(l) where l is Linear_Combination of B
    by VECTSP_7:def 2; then
  consider l being Linear_Combination of B1 such that
  C1: u = Sum(l);
  reconsider al = a * l as Linear_Combination of B1 by VECTSP_6:31;
  thus a * T.(u)
    = a * Sum(f (#) l) by A,C1
   .= Sum(f (#) al) by lemmult
   .= T.(Sum al) by A
   .= T.(a*u) by C1,VECTSP_6:45;
  end; then
reconsider T as linear-transformation of U,V by C,VECTSP_1:def 20,MOD_2:def 2;
B c= the carrier of U; then
B c= dom T by FUNCT_2:def 1; then
E: dom T /\ B = B by XBOOLE_1:28 .= dom f by FUNCT_2:def 1;
now let x be object;
  assume x in dom f; then
  reconsider w = x as Element of B;
  w in the carrier of (Lin B) by B; then
  w in the set of all Sum(l) where l is Linear_Combination of B
    by VECTSP_7:def 2; then
  consider l being Linear_Combination of B1 such that
  C1: w = Sum(l);
  thus T.x = Sum(f (#) l) by C1,A .= f.x by C1,lembas;
  end; then
T|B = f by E,FUNCT_1:46 .= (canLinTrans f)|B by defcl; then
T = canLinTrans f by canLinuni;
hence thesis by A;
end;
