Well-ordered series of ordinals

From: Noel Vaillant (vaillant_at_probability.net)
Date: 01/11/05 

Date: 11 Jan 2005 04:04:09 -0800

Given a family (ai) of ordinals indexed by a well-ordered set I.
Define the sum sum_i ai as the unique ordinal isomorphic to the
well-ordered set:

X = \/_{i in I} {i} x ai

where the good order on X is given by:

[(i,x) <= (j,y)] <-> [ (i<j) \/ ((i=j)/\(x <= y)) ]

I would like someone to confirm that provided I is isomorphic
to a limit ordinal, we have the equality:

sum_i ai = sup_{J<I} sum_{i in J} ai
where J<I indicates that J is a strict initial segment of I. 

--
Noel.

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