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 Th48:
  v9 = v & v <> v1 & v <> v2 implies Degree(v9, X) = Degree(v, X)
proof
  assume that
A1: v9 = v and
A2: v <> v1 and
A3: v <> v2;
  thus Degree(v9, X) = card Edges_In(v,X) + card Edges_Out(v9,X) by A1,A3,Th43
    .= Degree(v, X) by A1,A2,Th44;
end;
