
theorem Th29:
  for G1, G2 being _Graph, W1 being Walk of G1, W2 being Walk of G2
  for n being odd Nat st W1.vertexSeq() = W2.vertexSeq() holds W1.n = W2.n
proof
  let G1, G2 be _Graph, W1 be Walk of G1, W2 be Walk of G2;
  let n be odd Nat;
  assume A1: W1.vertexSeq() = W2.vertexSeq();
  then A2: len W1 = len W2 by Lm3;
  reconsider m = n as odd Element of NAT by ORDINAL1:def 12;
  per cases;
  suppose A3: n <= len W1;
    hence W1.n = W1.vertexAt(n) by GLIB_001:def 8
      .= W2.vertexSeq().((m+1) div 2) by A1, A3, GLIB_001:72
      .= W2.vertexAt(n) by A2, A3, GLIB_001:72
      .= W2.n by A2, A3, GLIB_001:def 8;
  end;
  suppose n > len W1;
    then not n in dom W1 & not n in dom W2 by A2, FINSEQ_3:25;
    then W1.n = {} & W2.n = {} by FUNCT_1:def 2;
    hence thesis;
  end;
end;
