reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem
  W2 is trivial implies W1.append(W2) = W1
proof
  assume W2 is trivial;
  then
A1: len W2 = 1 by Lm55;
  now
    per cases;
    suppose
      W1.last() = W2.first();
      then
A2:   W1.append(W2) = W1 ^' W2 by Def10;
      then
A3:   len W1.append(W2) + 1 = len W1 + 1 by A1,CARD_1:27,FINSEQ_6:139;
      for k being Nat st 1 <= k & k <= len W1.append(W2) holds W1.append(
      W2).k = W1.k
      by A2,A3,FINSEQ_6:140;
      hence thesis by A3,FINSEQ_1:14;
    end;
    suppose
      W1.last() <> W2.first();
      hence thesis by Def10;
    end;
  end;
  hence thesis;
end;
