
theorem
  for G2 being _Graph, V being Subset of the_Vertices_of G2
  for G1 being addLoops of G2, V, v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 & v1 in V holds
    v1.inDegree() = v2.inDegree() +` 1 & v1.outDegree() = v2.outDegree() +` 1 &
    v1.degree() = v2.degree() +` 2
proof
  let G2 be _Graph, V be Subset of the_Vertices_of G2;
  let G1 be addLoops of G2, V, v1 be Vertex of G1, v2 be Vertex of G2;
  assume v1 = v2 & v1 in V;
  then consider e being object such that
    A1: e DJoins v1,v1,G1 & not e in the_Edges_of G2 and
    A2: v1.edgesIn() = v2.edgesIn() \/ {e} and
    A3: v1.edgesOut() = v2.edgesOut() \/ {e} and
    v1.edgesInOut() = v2.edgesInOut() \/ {e} by Th42;
  not e in v2.edgesIn() by A1;
  hence A4: v1.inDegree() = v2.inDegree() +` card {e}
      by A2, CARD_2:35, ZFMISC_1:50
    .= v2.inDegree() +` 1 by CARD_1:30;
  not e in v2.edgesOut() by A1;
  hence v1.outDegree() = v2.outDegree() +` card {e}
      by A3, CARD_2:35, ZFMISC_1:50
    .= v2.outDegree() +` 1 by CARD_1:30;
  hence v1.degree()
     = v2.inDegree() +` (1 +` (v2.outDegree() +` 1)) by A4, CARD_2:19
    .= v2.inDegree() +` (v2.outDegree() +` (1 +` 1)) by CARD_2:19
    .= v2.degree() +` 2 by CARD_2:19;
end;
