reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p, p1, p2 for Path of G,
  vs, vs1, vs2 for FinSequence of the carrier of G,
  e, X for set,
  n, m for Nat;
reserve G for finite Graph,
  v for Vertex of G,
  c for Chain of G,
  vs for FinSequence of the carrier of G,
  X1, X2 for set;
reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p for Path of G,
  vs for FinSequence of the carrier of G,
  v9 for Vertex of AddNewEdge(v1, v2),
  p9 for Path of AddNewEdge(v1, v2),
  vs9 for FinSequence of the carrier of AddNewEdge(v1, v2);
reserve G for finite Graph,
  v, v1, v2 for Vertex of G,
  vs for FinSequence of the carrier of G,
  v9 for Vertex of AddNewEdge(v1, v2);

theorem Th49: :: CycVerDeg1
  for c being cyclic Path of G holds Degree(v, rng c) is even
proof
  let c be cyclic Path of G;
  per cases;
  suppose
    c is empty;
    then reconsider rc = rng c as empty set;
    Degree(v, rc)= 2 * 0;
    hence thesis;
  end;
  suppose
A1: c is non empty;
    consider vs being FinSequence of the carrier of G such that
A2: vs is_vertex_seq_of c by GRAPH_2:33;
    thus Degree(v, rng c) is even
    proof
      per cases;
      suppose
        v in rng vs;
        hence thesis by A2,Lm4;
      end;
      suppose
        not v in rng vs;
        then Degree(v, rng c) = 2 * 0 by A1,A2,Th32;
        hence thesis;
      end;
    end;
  end;
end;
