let a be Real; :: thesis: for V being RealLinearSpace
for u, v being VECTOR of V st {u,v} is linearly-independent & u <> v & a <> 0 holds
{u,(a * v)} is linearly-independent
let V be RealLinearSpace; :: thesis: for u, v being VECTOR of V st {u,v} is linearly-independent & u <> v & a <> 0 holds
{u,(a * v)} is linearly-independent
let u, v be VECTOR of V; :: thesis: ( {u,v} is linearly-independent & u <> v & a <> 0 implies {u,(a * v)} is linearly-independent )
assume that
A1: {u,v} is linearly-independent and
A2: u <> v and
A3: a <> 0 ; :: thesis: {u,(a * v)} is linearly-independent

now

let b, c be Real; :: thesis: ( (b * u) + (c * (a * v)) = 0. V implies ( b = 0 & c = 0 ) )
assume (b * u) + (c * (a * v)) = 0. V ; :: thesis: ( b = 0 & c = 0 )
then 0. V = (b * u) + ((c * a) * v) by RLVECT_1:def 9;
then ( b = 0 & c * a = 0 ) by A1, A2, RLVECT_3:14;
hence ( b = 0 & c = 0 ) by A3, XCMPLX_1:6; :: thesis: verum

end;

hence {u,(a * v)} is linearly-independent by RLVECT_3:14; :: thesis: verum