
theorem Th18:
  for G being _Graph
  for V being non empty one-to-one ManySortedSet of the_Vertices_of G
  for E being one-to-one ManySortedSet of the_Edges_of G
  for W1 being Walk of G ex W2 being Walk of replaceVerticesEdges(V,E)
  st V*(W1.vertexSeq()) = W2.vertexSeq() & E*(W1.edgeSeq()) = W2.edgeSeq()
proof
  let G be _Graph;
  let V be non empty one-to-one ManySortedSet of the_Vertices_of G;
  let E be one-to-one ManySortedSet of the_Edges_of G;
  consider F being PGraphMapping of G, replaceVerticesEdges(V,E) such that
    A1: F_V = V & F_E = E & F is Disomorphism by Th16;
  reconsider F as non empty PGraphMapping of G,replaceVerticesEdges(V,E) by A1;
  let W1 be Walk of G;
  reconsider W1 as F-defined Walk of G by A1, GLIB_010:121;
  take F.:W1;
  thus thesis by A1, GLIB_010:def 37;
end;
