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

now

let a, b be Real; :: thesis: ( (a * u) + (b * (v - u)) = 0. V implies ( a = 0 & b = 0 ) )
assume (a * u) + (b * (v - u)) = 0. V ; :: thesis: ( a = 0 & b = 0 )
then 0. V = (a * u) + ((b * v) - (b * u)) by RLVECT_1:48
.= ((a * u) + (b * v)) - (b * u) by RLVECT_1:def 6
.= ((a * u) - (b * u)) + (b * v) by RLVECT_1:def 6
.= ((a - b) * u) + (b * v) by RLVECT_1:49 ;
then ( a - b = 0 & b = 0 ) by A1, A2, RLVECT_3:14;
hence ( a = 0 & b = 0 ) ; :: thesis: verum

end;

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