Can you give an example of a scenario in which the President and the parliament have differed on policy and the PM has not acted in the interest of the President but of the National Assembly instead, trumping the President's power?  How would this work?  Why would the PM not represent the President's agenda?  Who has veto power, does the PM cast deciding votes, or is it mostly a position of influence and lobbying (when pres. has majority) and diplomacy (during "cohabitation")?

Also, I am confused again.  Upthread, afew states that the President chooses the PM, not the parliament:

In France, like in the US, the head of state is an elected president who does have a lot of political power. That, among others, of appointing the executive.
...
(and)
the French pres chooses a PM who draws up a list of ministers for approval (in fact the pres does most of the choosing).

and you say:

the executive power is in fact vested with the PM, chosen by the National Assembly.

Even during "cohabitation" the President still picks the PM, just from a different party.  Who has the final say on policies?  The President or the PM?  

Those who can make you believe absurdities can make you commit atrocities. -Voltaire