
theorem Th31:
  for G1, G2 being _Graph, W1 being Walk of G1, W2 being Walk of G2
  st W1.vertexSeq() = W2.vertexSeq() holds
    (W1 is non trivial implies W2 is non trivial) &
    (W1 is closed implies W2 is closed)
proof
  let G1, G2 be _Graph, W1 be Walk of G1, W2 be Walk of G2;
  assume A1: W1.vertexSeq() = W2.vertexSeq();
  hereby
    assume W1 is non trivial;
    then W1.length() <> 0 by GLIB_001:def 26;
    then W2.length() <> 0 by A1, Th30;
    hence W2 is non trivial by GLIB_001:def 26;
  end;
  hereby
    assume W1 is closed;
    then W1.first() = W1.last() by GLIB_001:def 24;
    then W1.first() = W2.last() by A1, Th30;
    then W2.first() = W2.last() by A1, Th30;
    hence W2 is closed by GLIB_001:def 24;
  end;
end;
