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 Th33:
  for W1A, W1B being Walk of G1, W2A,W2B being Walk of G2 st W1A =
  W2A & W1B = W2B holds W1A.append(W1B) = W2A.append(W2B)
proof
  let W1A, W1B be Walk of G1, W2A, W2B be Walk of G2;
  assume that
A1: W1A = W2A and
A2: W1B = W2B;
  now
    per cases;
    suppose
A3:   W1A.last() = W1B.first();
      then
A4:   W2A.last() = W2B.first() by A1,A2;
      thus W1A.append(W1B) = W1A ^' W1B by A3,Def10
        .= W2A.append(W2B) by A1,A2,A4,Def10;
    end;
    suppose
A5:   W1A.last() <> W1B.first();
      then
A6:   W2A.last() <> W2B.first() by A1,A2;
      thus W1A.append(W1B) = W2A by A1,A5,Def10
        .= W2A.append(W2B) by A6,Def10;
    end;
  end;
  hence thesis;
end;
