reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem
  W.first() = W.vertexSeq().1 & W.last() = W.vertexSeq().(len W .vertexSeq())
proof
A1: len W + 1 = 2*len W.vertexSeq() by Def14;
A2: 1 <= len W.vertexSeq() by Th65;
  then W.vertexSeq().1 = W.(2*1-1) by Def14;
  hence W.vertexSeq().1 = W.first();
  W.vertexSeq().(len W.vertexSeq()) = W.(2*len W.vertexSeq()-1) by A2,Def14;
  hence thesis by A1;
end;
