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 Th47:
  v9 = v & v1 <> v2 & (v = v1 or v = v2) & (the carrier' of G) in
  X implies Degree(v9, X) = Degree(v, X) +1
proof
  assume that
A1: v9 = v and
A2: v1 <> v2 and
A3: v = v1 or v = v2 and
A4: (the carrier' of G) in X;
  set E = the carrier' of G;
  per cases by A3;
  suppose
A5: v = v1;
    then
    Edges_In(v9, X) = Edges_In(v, X) & Edges_Out(v9, X) = Edges_Out(v, X)
    \/ {E} by A1,A2,A4,Th39,Th41;
    hence
    Degree(v9, X) = card Edges_In(v, X) + (card Edges_Out(v, X) + card {E
    }) by A1,A4,A5,Th41,CARD_2:40
      .= card Edges_In(v, X) + card Edges_Out(v, X) + card {E}
      .= Degree(v, X) +1 by CARD_1:30;
  end;
  suppose
A6: v = v2;
    then
    Edges_Out(v9, X) = Edges_Out(v, X) & Edges_In(v9, X) = Edges_In(v, X)
    \/ {E} by A1,A2,A4,Th40,Th42;
    hence
    Degree(v9, X) = card Edges_In(v, X) + card {E} + card Edges_Out(v, X)
    by A1,A4,A6,Th42,CARD_2:40
      .= card Edges_In(v, X) + card Edges_Out(v, X) + card {E}
      .= Degree(v, X) +1 by CARD_1:30;
  end;
end;
