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WOLOGWithout Loss of Generality (mathematics)
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Without loss of generality and for the clarity of illustration, only the effects of the clutters' Doppler frequencies are studied.
Assume without loss of generality that [[DELTA].sub.1] = [DELTA] and [T.sub.1] is a hanging tree on [v.sub.1].
Without loss of generality, we may assume that x(t) is eventually positive and bounded for all t [greater than or equal to] [t.sub.1] [greater than or equal to] [t.sub.0].
Thus, from (3.5) we have [[beta].sub.j] = 0, without loss of generality, we may assume j = 1.
Without loss of generality, suppose x(t) > 1 for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0.
Without loss of generality, assume that [v.sub.0] [v.sub.2] [not member of] E(G).
Without loss of generality, we can assume that there is an edge between [v.sub.k] and [v.sub.k + 1] but not between [u.sub.k] and [v.sub.k + 1].
Without loss of generality, it is assumed that [[theta].sup.i,i.sub.k] = 1.
If [absolute value of [C.sub.f'](u - v)] [less than or equal to] 2, then we can extend f' to a k'-GVDTC of [S.sub.m,n] by coloring vertices u, v and edge uv with any three different colors in [1,k]\{f'(u - v)}; if [absolute value of [C.sub.f'](u - v)] = l' [greater than or equal to] 3, we without loss of generality assume [C.sub.f'] (u - v)\ f'(u - v) = [1, l'].
Without loss of generality, let [C.sub.1] = {1,2,...,[r.sub.1]}, [C.sub.1] = {[r.sub.1] + 1,...,[r.sub.1] + [r.sub.2]},..., [C.sub.m] = {[r.sub.1] + ...
Without loss of generality assume that these components are the first m' ones: [u.sub.1], [u.sub.2], ...