
theorem
  for G1, G2 being _Graph, F being non empty one-to-one PGraphMapping of G1, G2
  for W1 being F-defined Walk of G1 st F_V.(W1.first()) = F_V.(W1.last())
  holds W1.first() = W1.last()
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;
  A1: F is semi-continuous;
  assume A2: F_V.(W1.first()) = F_V.(W1.last());
  per cases;
  suppose A3: len W1 >= 3;
    then 1 < len W1 by XXREAL_0:2;
    then A4: W1.(1+1) Joins W1.1, W1.(1+2), G1 by POLYFORM:4, GLIB_001:def 3;
    A5: W1.1 in W1.vertices() & W1.3 in W1.vertices()
      & W1.last() in W1.vertices()
      by A3, XXREAL_0:2, GLIB_001:87, GLIB_001:88, POLYFORM:4, POLYFORM:6;
    1 <= 2 & 2 <= len W1 by A3, XXREAL_0:2;
    then W1.2 in W1.edges() by POLYFORM:5, GLIB_001:99;
    then A6: W1.1 in dom F_V & W1.2 in dom F_E & W1.3 in dom F_V
      & W1.last() in dom F_V by A5, Def35, TARSKI:def 3;
    F_E.(W1.2) Joins F_V.(W1.1), F_V.(W1.3), G2 by A6, A4, Th4;
    then F_E.(W1.2) Joins F_V.(W1.last()), F_V.(W1.3), G2
      by A2, GLIB_001:def 6;
    then W1.2 Joins W1.last(), W1.3, G1 by A1, A6;
    then (W1.1 = W1.last() & W1.3 = W1.3) or (W1.1 = W1.3 & W1.3 = W1.last())
      by A4, GLIB_000:15;
    hence thesis by GLIB_001:def 6;
  end;
  suppose not (len W1 >= 3);
    hence thesis by GLIB_001:125, GLIB_001:127;
  end;
end;
