
theorem
for F being Field,
    E being FieldExtension of F,
    K being F-extending FieldExtension of E
for BE being linearly-independent Subset of VecSp(E,F),
    BK being linearly-independent Subset of VecSp(K,E)
for a1,a2,b1,b2 being Element of K
st a1 in BE & a2 in BE & b1 in BK & b2 in BK
holds a1 * b1 = a2 * b2 implies (a1 = a2 & b1 = b2)
proof
let F be Field, E be FieldExtension of F,
    K be F-extending FieldExtension of E;
let BE be linearly-independent Subset of VecSp(E,F),
    BK be linearly-independent Subset of VecSp(K,E);
let a1, a2, b1, b2 be Element of K;
assume A1: a1 in BE & a2 in BE & b1 in BK & b2 in BK;
then reconsider b1v = b1 as Element of VecSp(K,E);
assume A2: a1 * b1 = a2 * b2;
{b1v} c= BK by A1,TARSKI:def 1;
then b1v <> 0.VecSp(K,E) by VECTSP_7:1,VECTSP_7:3;
then A3: b1 <> 0.K by FIELD_4:def 6;
A4: b1 = b2 by A1,A2,BE0;
a1 * b1 = b2 * a2 by A2,GROUP_1:def 12;
then b1 * a1 = b2 * a2 by GROUP_1:def 12;
hence thesis by A3,A4,VECTSP_2:8;
end;
