The Euler circuits of the graphs Gi are \[\eqalign{ &w_{1,1},w_{1,2},\ldots,w_{1,m_1}=w_{1,1}\cr &w_{2,1},w_{2,2},\ldots,w_{2,m_2}=w_{2,1}\cr &\vdots\cr &w_{k,1},w_{k,2},\ldots,w_{k,m_k}=w_{k,1}....The Euler circuits of the graphs Gi are w1,1,w1,2,…,w1,m1=w1,1w2,1,w2,2,…,w2,m2=w2,1⋮wk,1,wk,2,…,wk,mk=wk,1. By pasting together the original closed walk with these, we form a closed walk in G that uses every edge exactly once: \[\eqalign{ v_0,v_1,&\ldots,v_{i_1}=w_{1,1},w_{1,2},\ldots,w_{1,m_1}=v_{i_1},v_{i_1+1},\cr &\ldots, v_{i_2}=w_{2,1},\ldots,w_{2,m_2}=v_{i_2},v_{i_2+1},\cr &\ld…
