
theorem XXX2:
for F being Field
for U,V being VectSp of F
for B being non empty finite Subset of U
for f being Function of B,V st f is one-to-one
for l being Linear_Combination of B
for v being Element of V st v in rng f holds (f (#) l).v = l.(f".v)
proof
let F be Field, U,V be VectSp of F;
let B be non empty finite Subset of U, f be Function of B,V;
assume A: f is one-to-one;
let l be Linear_Combination of B, v be Element of V;
assume B: v in rng f; then
consider x being object such that C: x in dom f & f.x = v by FUNCT_1:def 3;
I: dom f = B by FUNCT_2:def 1; then
reconsider u = x as Element of U by C;
H: dom l = the carrier of U by FUNCT_2:def 1;
v in dom(f") by A,B,FUNCT_1:33; then
f".v in rng(f") by FUNCT_1:3; then
K: f".v in dom f by A,FUNCT_1:33; then
J: f".v in U by I;
D: l.((f").v) in rng l by K,I,H,FUNCT_1:3;
G: f"{v} = {(f").v} by A,B,XXX1; then
E: canFS f"{v}  = <* (f").v *> by FINSEQ_1:94;
set G = l * (canFS f"{v});
F: dom G = Seg 1
   proof
   F1: now let o be object;
       assume o in Seg 1; then
       F2: o in dom canFS f"{v} by E,FINSEQ_1:38; then
       (canFS f"{v}).o in rng(canFS f"{v}) by FUNCT_1:3; then
       (canFS f"{v}).o = f".v by G,TARSKI:def 1; then
       (canFS f"{v}).o in dom l by J,FUNCT_2:def 1;
       hence o in dom G by F2,FUNCT_1:11;
       end;
   now let o be object;
       assume o in dom G; then
       o in dom canFS f"{v} by FUNCT_1:11;
       hence o in Seg 1 by E,FINSEQ_1:38;
       end;
   hence thesis by F1,TARSKI:2;
   end; then
reconsider G as FinSequence by FINSEQ_1:def 2;
now let o be object;
  assume o in rng G; then
  o in rng l by FUNCT_1:14;
  hence o in the carrier of F;
  end; then
rng G c= the carrier of F; then
reconsider G as FinSequence of the carrier of F by FINSEQ_1:def 4;
dom canFS f"{v} = Seg 1 by E,FINSEQ_1:38; then
1 in dom canFS f"{v}; then
G.1 = l.((canFS f"{v}).1) by FUNCT_1:13
    .= l.((f").v) by E; then
l * (canFS f"{v}) = <* l.((f").v) *> by F,FINSEQ_1:def 8; then
Sum Expand(f,l,v) = l.((f").v) by D,RLVECT_1:44;
hence thesis by defK;
end;
