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;
