Maybe I'm reading this wrong, but if it's anything like the Mincome Experiment (http://en.wikipedia.org/wiki/Mincome) then it means you get $2800 unconditionally, but a $1200 a month job comes out of it.
The way these programs are paid for is because most people would be making more than $2800 a month anyway, so their net cost to the government is zero.
Mincome != Basic Income. Under Basic Income, everyone gets a fixed amount unconditionally. Any extra income you get on top of that is yours to keep. So you get the $2800 from the government, then get a job for that $1200 (from the grandparent's example), so now you have $4000 in total.
Mincome worked differently from this, but the OP talks about Basic Income, which just gives a fixed amount to everyone.
Actually, now that I'm doing more reading, things can be a bit more confusing.
There are a few approaches to solve this problem - in some cases a negative income tax is applied, and in others the government actually issues a 'citizens dividend'.
Neither case necessarily mandates a 1:1 decrease in benefit as your other income sources go up. It all depends on how things are structured... someone with a significant monthly income could have a high enough tax as to effectively remove the base payment, where a particularly poor worker may not lose any.
And unfortunately I don't speak enough French to read the petition, and none of the sources in English I can find actually spell out the details.
The way these programs are paid for is because most people would be making more than $2800 a month anyway, so their net cost to the government is zero.