
theorem
  for G2 being _Graph, V being set, G1 being addVertices of G2, V
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 holds
    v1.edgesIn() = v2.edgesIn() & v1.inDegree() = v2.inDegree() &
    v1.edgesOut() = v2.edgesOut() & v1.outDegree() = v2.outDegree() &
    v1.edgesInOut() = v2.edgesInOut() & v1.degree() = v2.degree()
proof
  let G2 be _Graph, V be set, G1 be addVertices of G2, V;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  A2: G2 is Subgraph of G1 by GLIB_006:57;
  then A3: v2.edgesIn() c= v1.edgesIn() by A1, GLIB_000:78;
  now
    let e be object;
    assume e in v1.edgesIn();
    then consider x being set such that
      A4: e DJoins x,v1,G1 by GLIB_000:57;
    e DJoins x,v2,G2 & e is set by A1, A4, GLIB_006:85, TARSKI:1;
    hence e in v2.edgesIn() by GLIB_000:57;
  end;
  then v1.edgesIn() c= v2.edgesIn() by TARSKI:def 3;
  hence A5: v1.edgesIn() = v2.edgesIn() by A3, XBOOLE_0:def 10;
  hence A6: v1.inDegree() = v2.inDegree();
  A7: v2.edgesOut() c= v1.edgesOut() by A1, A2, GLIB_000:78;
  now
    let e be object;
    assume e in v1.edgesOut();
    then consider x being set such that
      A8: e DJoins v1,x,G1 by GLIB_000:59;
    e DJoins v2,x,G2 & e is set by A1, A8, GLIB_006:85, TARSKI:1;
    hence e in v2.edgesOut() by GLIB_000:59;
  end;
  then v1.edgesOut() c= v2.edgesOut() by TARSKI:def 3;
  hence A9: v1.edgesOut() = v2.edgesOut() by A7, XBOOLE_0:def 10;
  hence A10: v1.outDegree() = v2.outDegree();
  thus v1.edgesInOut() = v2.edgesInOut() by A5, A9;
  thus v1.degree() = v2.degree() by A6, A10;
end;
