
theorem
  for G1, G2 being _Graph
  for F being non empty one-to-one PGraphMapping of G1, G2
  for W2 being F-valued Walk of G2 holds F.:(F"W2) = W2
proof
  let G1, G2 be _Graph, F be non empty one-to-one PGraphMapping of G1, G2;
  let W2 be F-valued Walk of G2;
  W2.vertices() c= rng F_V by Def36;
  then A1: rng W2.vertexSeq() c= rng F_V by GLIB_001:def 16;
  A2: (F.:(F"W2)).vertexSeq() = F_V * (F".:W2).vertexSeq() by Def37
    .= F_V * (F"_V * W2.vertexSeq()) by Def37
    .= (F_V * F_V") * W2.vertexSeq() by RELAT_1:36
    .= (id rng F_V) * W2.vertexSeq() by FUNCT_1:39
    .= W2.vertexSeq() by A1, RELAT_1:53;
  W2.edges() c= rng F_E by Def36;
  then A3: rng W2.edgeSeq() c= rng F_E by GLIB_001:def 17;
  (F.:(F"W2)).edgeSeq() = F_E * (F".:W2).edgeSeq() by Def37
    .= F_E * (F"_E * W2.edgeSeq()) by Def37
    .= (F_E * F_E") * W2.edgeSeq() by RELAT_1:36
    .= (id rng F_E) * W2.edgeSeq() by FUNCT_1:39
    .= W2.edgeSeq() by A3, RELAT_1:53;
  hence thesis by A2, GLIB_009:26;
end;
