[sukilot]

"Torus is product of two circles" is a basic fact of algebra/geometry/analysis. You can easily assign a coordinate system [0,2pi] x [0-2pi], with 0=2pi.

But I guess it's cool you can prove the same thing with over-abstract gobbledygook

[lmkg]

When you're dealing with a totally new field, there's a lot of low-level, seemingly-obvious junk to sort out. And immature fields haven't developed common tools or proof techniques, so everything looks harder than it should be. Give it some time, the field might get more approachable, expressive, and useful as it matures.

[theOnliest]

I've always found these images [1,2] from Wikipedia to be helpful in explaining how the torus is a product of two circles.

[1] http://en.wikipedia.org/wiki/Torus#mediaviewer/File:Sphere-l... [2] http://en.wikipedia.org/wiki/Torus#mediaviewer/File:Torus_cy...