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As paths p1 and p2 are disconnected, there must be at least two components g1 and g2.

As g1 and g2 are disconnected there is no vertex in common and no vertex in g1 has a path to any vertex in g2.

Chose any two vertext v1 and v2 where v1 is a vertex in g1 and v2 in g2.

Now path p1 must contain these two vertices ..so there is a path between them in p1 with some intermediate vertice in p1.

Let this path be v1,a,b,c,...,v2

Similarly we have for p2, a path between v1 and v2 i.e; v1, r,s,t,...,v2.

Now the set {a,b,c,...} and {r,s,t,...} are mutually exclusive .

When we take the union of paths p1 and p2 as P , we include the vertex set {a,b,c,...} and {r,s,t,...} in P.

We can easily see that v1,a,b,c,...,v2,...,t,s,r,v1 forms a circuit.

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