
theorem Th123:
  for G1, G2 being _Graph
  for F being non empty one-to-one PGraphMapping of G1, G2
  for W1 being F-defined Walk of G1 holds F"(F.:W1) = W1
proof
  let G1, G2 be _Graph, F be non empty one-to-one PGraphMapping of G1, G2;
  let W1 be F-defined Walk of G1;
  W1.vertices() c= dom F_V by Def35;
  then A1: rng W1.vertexSeq() c= dom F_V by GLIB_001:def 16;
  A2: (F"(F.:W1)).vertexSeq() = F"_V * ((F.:W1).vertexSeq()) by Def37
    .= F_V" * (F_V * W1.vertexSeq()) by Def37
    .= (F_V" * F_V) * W1.vertexSeq() by RELAT_1:36
    .= (id dom F_V) * W1.vertexSeq() by FUNCT_1:39
    .= W1.vertexSeq() by A1, RELAT_1:53;
  W1.edges() c= dom F_E by Def35;
  then A3: rng W1.edgeSeq() c= dom F_E by GLIB_001:def 17;
  (F"(F.:W1)).edgeSeq() = F"_E * ((F.:W1).edgeSeq()) by Def37
    .= F_E" * (F_E * W1.edgeSeq()) by Def37
    .= (F_E" * F_E) * W1.edgeSeq() by RELAT_1:36
    .= (id dom F_E) * W1.edgeSeq() by FUNCT_1:39
    .= W1.edgeSeq() by A3, RELAT_1:53;
  hence thesis by A2, GLIB_009:26;
end;
