If $M_1$ and $M_2$ are well ordered then $ M = M_1 \times M_2$ is well ordered too.
with the orders $$ (a_1,b_1) \leq (a_2,b_2) \ iff \ b_1 \leq b_2 \ in \ M_2 \ and \ a_1 \leq a_2 \ in \ M_1 \ ,\ b_1=b_2$$
I prove that this is a total order, I prove trichotomy, transitive, but how can I prove that there existe a first element in M.
I tried this by contradiction if $(a_0,b_0) \in M$ is first element but I don't know how continue this idea.
Someone can help me please. Thanks for your time and help.