When is an OLS parameter unchanged on a subsample? There is a sample of $n$ observations, each element has a numeric $Y$ and $X$ characteristic. There is an OLS regression over the sample
$$
Y = b_0 + b_1 X + \textbf{u},
$$
$\textbf{u}$ being the vector of residuals. Suppose we remove observation $i$ from the sample, restricting it to $n-1$ observations, and run an OLS regression again, yielding
$$
Y_{-i} = b_0' + b_1' X_{-i} + \textbf{u}'.
$$
Subscript $_{-i}$ denotes that $Y$ and $X$ are not the same vectors as before, as observation $i$ is missing.

Seems to me that if the observation $i$ we removed is on the original sample's 'regression line', that is if
$$
Y_i = b_0 + b_1 X_i,
$$
then $b_0 = b_0'$ and $b_1 = b_1'$.

Example: (in R code)

x = c(5,3,4,5,4,4)
y = c(20,15,14,21,10,25)

plot(x,y)
abline(coef(reg))




plot(x[-1],y[-1])

reg1 = lm(y[-1] ~ x[-1])
abline(coef(reg1))




My questions are:

1) Is this true, and if yes, what is the proof? 
[I figured this one out in the meantime, but feel free to give a nicer solution.]

2) Is it possible that after removing an observation (not on the 'regression line') we have

($b_0 = b_0'$ and $b_1 \neq b_1'$)  or ($b_0 \neq b_0'$ and $b_1 = b_1'$)?
I think I figured out 1). I am still interested in 2).

OLS sets $b_0$ and $b_1$ in such a way that

$$
(b_0 \ \  b_1) = \arg\min_{a_0, a_1} \sum_j \left(Y_j - a_0 - a_1 X_j\right)^2. 
$$
The first order conditions are
$$
(-2)\sum_j \left(Y_j - b_0 - b_1 X_j\right) = 0 
$$
and
$$
(-2)\sum_j X_j \left(Y_j - b_0 - b_1 X_j\right) = 0.
$$
If observation $i$ is on the regression line then
$$
\begin{align*}
\left(Y_i - b_0 - b_1 X_i\right) &amp; = 0 \\
\\
X_i\left(Y_i - b_0 - b_1 X_i\right) &amp; = 0.
\end{align*}
$$
It follows from this and the original first order conditions that
$$
\begin{align*}
(-2)\sum_{j\neq i} \left(Y_j - b_0 - b_1 X_j\right) &amp; = 0 \\
\\
(-2)\sum_{j\neq i} X_j\left(Y_j - b_0 - b_1 X_j\right) &amp; = 0,
\end{align*}
$$
which are the first order conditions of the subsample's OLS problem.