
theorem XXX4:
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 (rng f) 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 (rng f);
set l2 = l (#) f;
H: dom f = B by FUNCT_2:def 1;
now let v be Element of V;
  per cases;
  suppose C: v in rng f; then
    consider u being object such that
    D: u in dom f & f.u = v by FUNCT_1:def 3;
    reconsider u as Element of U by H,D;
    E: f".v = u by D,A,FUNCT_1:34;
    l2.u = l.(f.u) by D,defK1;
    hence (f (#) l2).v = l.v by D,E,C,A,XXX2;
    end;
  suppose C: not v in rng f;
    Carrier l c= rng f by VECTSP_6:def 4; then
    not v in Carrier l by C; then
    D: l.v = 0.F;
    Carrier(f (#) l2) c= rng f by VECTSP_6:def 4; then
    not v in Carrier(f (#) l2) by C;
    hence (f (#) l2).v = l.v by D;
    end;
  end;
hence thesis by VECTSP_6:def 7;
end;
