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
  for G1 being _Graph, G2 being Subgraph of G1, W1 being Walk of G1, W2
being Walk of G2, e being set st W1 = W2 & e in W2.last().edgesInOut() holds W1
  .addEdge(e) = W2.addEdge(e)
proof
  let G1 be _Graph, G2 be Subgraph of G1, W1 be Walk of G1, W2 be Walk of G2,
  e be set;
  assume that
A1: W1 = W2 and
A2: e in W2.last().edgesInOut();
  set W2B = G2.walkOf(W2.last(), e, W2.last().adj(e));
  set W1B = G1.walkOf(W1.last(), e, W1.last().adj(e));
A3: e Joins W2.last(),W2.last().adj(e),G2 by A2,GLIB_000:67;
  W1.last().adj(e) = W2.last().adj(e) by A1,A2,GLIB_000:80;
  then W1B = W2B by A1,A3,Th171;
  hence thesis by A1,Th33;
end;
