
theorem
  for G2 being _Graph, V being set, G1 being addVertices of G2, V
  holds
    G1.supDegree() = G2.supDegree() &
    G1.supInDegree() = G2.supInDegree() &
    G1.supOutDegree() = G2.supOutDegree()
proof
  let G2 be _Graph, V be set, G1 be addVertices of G2, V;
  G2 is Subgraph of G1 by GLIB_006:57;
  then A1: G2.supDegree() c= G1.supDegree() &
    G2.supInDegree() c= G1.supInDegree() &
    G2.supOutDegree() c= G1.supOutDegree() by Th63;
  A2: the_Vertices_of G1 = the_Vertices_of G2 \/ V by GLIB_006:def 10;
  set D1 = the set of all v.degree() where v is Vertex of G1;
  set D2 = the set of all v.degree() where v is Vertex of G2;
  now
    let x be object;
    assume x in G1.supDegree();
    then consider d1 being set such that
      A3: x in d1 & d1 in D1 by TARSKI:def 4;
    consider v1 being Vertex of G1 such that
      A4: d1 = v1.degree() by A3;
    not v1 in V \ the_Vertices_of G2 by A3, A4, GLIB_000:157, GLIB_006:88;
    then v1 in the_Vertices_of G2 or not v1 in V by XBOOLE_0:def 5;
    then reconsider v2 = v1 as Vertex of G2 by A2, XBOOLE_0:def 3;
    d1 = v2.degree() & v2.degree() in D2 by A4, GLIBPRE0:45;
    hence x in G2.supDegree() by A3, TARSKI:def 4;
  end;
  then G1.supDegree() c= G2.supDegree() by TARSKI:def 3;
  hence G1.supDegree() = G2.supDegree() by A1, XBOOLE_0:def 10;
  set D3 = the set of all v.inDegree() where v is Vertex of G1;
  set D4 = the set of all v.inDegree() where v is Vertex of G2;
  now
    let x be object;
    assume x in G1.supInDegree();
    then consider d1 being set such that
      A5: x in d1 & d1 in D3 by TARSKI:def 4;
    consider v1 being Vertex of G1 such that
      A6: d1 = v1.inDegree() by A5;
    v1 is non isolated by A5, A6, GLIB_000:156;
    then not v1 in V \ the_Vertices_of G2 by GLIB_006:88;
    then v1 in the_Vertices_of G2 or not v1 in V by XBOOLE_0:def 5;
    then reconsider v2 = v1 as Vertex of G2 by A2, XBOOLE_0:def 3;
    d1 = v2.inDegree() & v2.inDegree() in D4 by A6, GLIBPRE0:45;
    hence x in G2.supInDegree() by A5, TARSKI:def 4;
  end;
  then G1.supInDegree() c= G2.supInDegree() by TARSKI:def 3;
  hence G1.supInDegree() = G2.supInDegree() by A1, XBOOLE_0:def 10;
  set D5 = the set of all v.outDegree() where v is Vertex of G1;
  set D6 = the set of all v.outDegree() where v is Vertex of G2;
  now
    let x be object;
    assume x in G1.supOutDegree();
    then consider d1 being set such that
      A7: x in d1 & d1 in D5 by TARSKI:def 4;
    consider v1 being Vertex of G1 such that
      A8: d1 = v1.outDegree() by A7;
    v1 is non isolated by A7, A8, GLIB_000:156;
    then not v1 in V \ the_Vertices_of G2 by GLIB_006:88;
    then v1 in the_Vertices_of G2 or not v1 in V by XBOOLE_0:def 5;
    then reconsider v2 = v1 as Vertex of G2 by A2, XBOOLE_0:def 3;
    d1 = v2.outDegree() & v2.outDegree() in D6 by A8, GLIBPRE0:45;
    hence x in G2.supOutDegree() by A7, TARSKI:def 4;
  end;
  then G1.supOutDegree() c= G2.supOutDegree() by TARSKI:def 3;
  hence G1.supOutDegree() = G2.supOutDegree() by A1, XBOOLE_0:def 10;
end;
