
theorem Th16:
  for G being _Graph, W1,W2 being Walk of G st W1 is trivial & W1
  .last() = W2.first() holds W1.append(W2) = W2
proof
  let G be _Graph, W1,W2 be Walk of G such that
A1: W1 is trivial and
A2: W1.last() = W2.first();
A3: len W1 = 1 by A1,GLIB_001:126;
  then
A4: len W1.append(W2) + 1 = 1 + len W2 by A2,GLIB_001:28;
  now
    let k be Nat such that
A5: 1 <= k and
A6: k <= len W1.append(W2);
    1-1 <= k-1 by A5,XREAL_1:9;
    then reconsider k1 = k-1 as Element of NAT by INT_1:3;
    k-1 < k by XREAL_1:44;
    then k-1 < len W2 by A4,A6,XXREAL_0:2;
    then W1.append(W2).(1 + k1) = W2.(k1+1) by A2,A3,GLIB_001:33;
    hence W1.append(W2).k = W2.k;
  end;
  hence thesis by A4,FINSEQ_1:14;
end;
