Just work from the definitions and find a path that satisfies the conditions.

1: You can take the vertice sequence {x} which starts and stops at x and exists by axiom.

2: if F: (a,b,...,z) is the path from x to y, then G: (z,...,b,a) is the path from y to x, which exists because every edge q in G exists by assumption since it exists in F.

3: if F: (a,b,...,m), G: (n,o...,z) are the respective paths from x to y and from y to z, then the path Ha,b,...,m,n,o,...,z) is the path from x to z, and all edges exist because they existed in F and G.

I couldn't be arsed with indices of the vertices, which you should use if you do this for homework, but using the alphabet is clearer.

General rule of thumb when solving these things: don't think too deeply and just construct the answers straight from the definition and question.