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 A1: ( {u,v} is linearly-independent & 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 A2: 0. V = (a * u) + ((b * u) + (b * v)) by RLVECT_1:def 8
.= ((a * u) + (b * u)) + (b * v) by RLVECT_1:def 6
.= ((a + b) * u) + (b * v) by RLVECT_1:def 9 ;
then b = 0 by A1, RLVECT_3:14;
hence ( a = 0 & b = 0 ) by A1, A2, RLVECT_3:14; :: thesis: verum

end;

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