understanding a proof

[jaunty pecan]

mid

[radiant raven]

hey can't understand a part of a proof can someone enlighten it? don't understand the last implication

I have ${v_1,...,v_n}$ and ${w_1,...,w_n}$ two orthogonal basis of $V$ a vectorial space on $\mathbb{R}$ with a symetric bilinear form $\langle,\rangle$.

$\langle v_i,v_i\rangle > 0$ if $1\leq i\leq r$, $\langle v_i,v_i\rangle < 0$ if $r+1\leq i\leq s$, $\langle v_i,v_i\rangle = 0$ if $s+1\leq i\leq n$, and the same for $w_i$ with $r', s'$.

i want to show that $v_1,...,v_r,w_{r'+1},...,w_n$ is linearly independant. By absurd suppose linearly dependant so:
[
x_1v_1 + \dots + x_rv_r=y_{r'+1}w_{r'+1} + y_nw_n
]
with the scalars $x_i$ and $y_i$ not all null.
which implies apparently:
[
x_1^2\langle v_1,v_1\rangle + \dots + x_r^2\langle v_r,vr\rangle = y_{r'+1}^2\langle w_{r'+1},w_{r'+1}\rangle + \dots + y_n^2\langle w_n, w_n\rangle
]