. ?i-?-{1 and . 2}-:-?x-?-l-i,

Y. ?x and . ?l,

, Proof. 1. By construction

, Let x, y ?L be two distinct elements such that x ? y. From the definition of ? V , it follows that ?z ? I : ? 1 (z) = x and ? 2 (z) = y (up to swapping of x and y)

. Let-x-?-y-then-?i-?-{1 and . 2}-:-x,-y-?-l-i, Analogously, ?j ? {1, 2} :x,? ? L j , and? < jx . Clearly it must be that i = j. Now, x ? Vx , and y ? V? ? ?a, b ? I : ? i (a) = x, ? j (a) =x, ? i (a) = y, and ? j (b) =?. Since ? i is an omp-embedding, it must reflect the order, yielding a ? b

, These results allow for endowing the quotientL/ ? V with a structure

, Consider the setting of Definition 11, and define: 1. L =L/ ? V , 2. 0 =

. ?-?-l-×-l,

, Then the I-pasting of L 1 and L 2 induced by V is the structure L 1 |I|L 2 = L, ?, p.1

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