
theorem Th21:
  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 W2 being Walk of replaceVerticesEdges(V,E) ex W1 being Walk of G
  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 one-to-one PGraphMapping of
    G,replaceVerticesEdges(V,E) by A1;
  let W2 be Walk of replaceVerticesEdges(V,E);
  reconsider W2 as F-valued Walk of replaceVerticesEdges(V,E)
    by A1, GLIB_010:122;
  take F"W2;
  thus thesis by A1, GLIB_010:def 39;
end;
