why not just start with a < b and then add c to both sides?

i'm not sure what you want here. these seem pretty straight forward. we have a < b < c. it follows by the transitivity of "<" that a < c.2) Let a,b,c e Z. If a < b and b < c, then a < c.

ANy advice?

if you want to do it more flashy, perhaps...

assume a < b and b < c. from a < b we have a - b < 0. but a - c < a - b since b < c. but that means we have a - c < 0 by transitivity of "<". now add c to both sides.

in any case, you would do well to tell us what axioms you have to work with