
theorem lembas:
for F being Field
for U,V being VectSp of F
for B being non empty finite Subset of U st B is linearly-independent
for w being Element of U st w in B
for l being Linear_Combination of B st Sum l = w
for f being Function of B,V holds Sum(f (#) l) = f.w
proof
let F be Field, U,V be VectSp of F, B be non empty finite Subset of U;
assume AS1: B is linearly-independent;
let w be Element of U;
assume AS2: w in B;
let l be Linear_Combination of B;
assume AS3: Sum l = w;
let f be Function of B,V;
set G = f (#) l;
reconsider b = w as Element of B by AS2;
I: Carrier l = { w } & l.w = 1.F by AS1,AS2,AS3,lembas1; then
Sum(f (#) l) = l.b * f.b by lemadd2;
hence thesis by I;
end;
