
theorem lemminusx:
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
for l1,l2,l3 being Linear_Combination of B
st l3 = l1 - l2 holds f (#) l3 = (f (#) l1) - (f (#) l2)
proof
let F be Field, U,V be VectSp of F, B be non empty finite Subset of U;
let f be Function of B,V; let l1,l2,l3 be Linear_Combination of B;
assume A: l3 = l1 - l2;
reconsider l4 = -l2 as Linear_Combination of B by VECTSP_6:39;
B: - (f (#) l2) = f (#) l4 by lemmultx;
thus f (#) l3 = (f (#) l1) - (f (#) l2) by A,B,lemaddx;
end;
