On 10/13/2013 8:13 AM, Hetware wrote: > > That's nonsense. In variational calculus we assert the existence of an > infinite number of continuous, twice differentiable functions which > coincide at the boundaries of the parameter domain, but are otherwise > arbitrary. > > A statement such as "let M be a smooth manifold", asserts the existence > of a continuous function. >
As Peter noted, you do not assert the properties of a defined function.
First, you check that it is well-defined. Then, having proven that the definition does, indeed, define a function (You do not even know that from a mere definition.), you prove that it does or does not satisfy particular properties.
The statement you have cited is a hypothetical. In a proof statement, there will be some explicit or implicit context by which
"If M is a smooth manifold, then..."
is in force.
Relative to this context, the premise is assumed to be true:
"Let M be a smooth manifold."
What is being done here is that the premise of an implication is being assumed to prove that the hypothetical expressed by the implication holds.