
theorem XXX3:
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 holds (f (#) l) (#) f = l
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;
set l2 = f (#) l;
H: dom f = B by FUNCT_2:def 1;
now let u be Element of U;
  per cases;
  suppose C: u in B; then
    f.u in rng f by H,FUNCT_1:3; then
    reconsider v = f.u as Element of V;
    l2.v = l.(f".v) by A,C,H,FUNCT_1:3,XXX2;
    hence (l2 (#) f).u = l.(f".v) by C,defK1 .= l.u by A,C,H,FUNCT_1:34;
    end;
  suppose C: not u in B;
    Carrier l c= B by VECTSP_6:def 4; then
    not u in Carrier l by C; then
    D: l.u = 0.F;
    Carrier(l2 (#) f) c= B by VECTSP_6:def 4;then
    not u in Carrier(l2 (#) f) by C;
    hence (l2 (#) f).u = l.u by D;
    end;
  end;
hence thesis by VECTSP_6:def 7;
end;
