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 Th68:
 for e,x,y being object holds
  e Joins x,y,G implies G.walkOf(x,e,y).vertexSeq() = <*x,y*>
proof let e,x,y be object;
  set W = G.walkOf(x,e,y);
  assume e Joins x, y, G;
  then
A1: W = <*x, e, y*> by Def5;
  len W + 1 = 2 * len W.vertexSeq() by Def14;
  then
A2: 3 + 1 = 2 * len W.vertexSeq() by A1,FINSEQ_1:45;
  then W.vertexSeq().2 = W.(2*2-1) by Def14;
  then
A3: W.vertexSeq().2 = y by A1;
  W.vertexSeq().1 = W.(2*1-1) by A2,Def14;
  then W.vertexSeq().1 = x by A1;
  hence thesis by A2,A3,FINSEQ_1:44;
end;
