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;

theorem Th31:
  Degree(v, X) = Degree(v, X/\the carrier' of G)
proof
  set E = the carrier' of G;
  thus Degree(v, X) = card Edges_In(v, X/\E) + card Edges_Out(v, X) by Th30
    .= Degree(v, X/\the carrier' of G) by Th30;
end;
