
theorem Th30:
  for G1, G2 being _Graph, W1 being Walk of G1, W2 being Walk of G2
  st W1.vertexSeq() = W2.vertexSeq() holds
    len W1 = len W2 & W1.length() = W2.length() &
    W1.first() = W2.first() & W1.last() = W2.last() &
    W2 is_Walk_from W1.first(),W1.last()
proof
  let G1, G2 be _Graph, W1 be Walk of G1, W2 be Walk of G2;
  assume A1: W1.vertexSeq() = W2.vertexSeq();
  hence len W1 = len W2 by Lm3;
  hence W1.length() = W2.length() by Lm2;
  thus A2: W1.first() = W1.1 by GLIB_001:def 6
    .= W2.1 by A1, Th29, POLYFORM:4
    .= W2.first() by GLIB_001:def 6;
  thus W1.last() = W1.len W1 by GLIB_001:def 7
    .= W1.len W2 by A1, Lm3
    .= W2.len W2 by A1, Th29
    .= W2.last() by GLIB_001:def 7;
  hence thesis by A2, GLIB_001:def 23;
end;
