
theorem
  for G being _Graph, C being Component of G
  for v1 being Vertex of G, v2 being Vertex of C 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 G be _Graph, C be Component of G;
  let v1 be Vertex of G, v2 be Vertex of C;
  assume A1: v1 = v2;
  then A2: v2.edgesIn() c= v1.edgesIn() & v2.edgesOut() c= v1.edgesOut()
    by GLIB_000:78;
  now
    let e be object;
    assume e in v1.edgesIn();
    then consider x being set such that
      A3: e DJoins x,v1,G by GLIB_000:57;
    set W = G.walkOf(v1,e,x);
    A4: e Joins v1,x,G by A3, GLIB_000:16;
    then W is_Walk_from v1,x by GLIB_001:15;
    then A5: x in G.reachableFrom(v1) by GLIB_002:def 5;
    A6: e Joins v2,x,G & e DJoins x,v2,G by A1, A3, A4;
    the_Vertices_of C = G.reachableFrom(v1) by A1, GLIB_002:33;
    then e in G.edgesBetween(the_Vertices_of C) by A5, A6, GLIB_000:32;
    then e in the_Edges_of C by GLIB_002:31;
    then e DJoins x,v2,C & e is set by A6, GLIB_000:73;
    hence e in v2.edgesIn() by GLIB_000:57;
  end;
  then v1.edgesIn() c= v2.edgesIn() by TARSKI:def 3;
  hence A7: v1.edgesIn() = v2.edgesIn() by A2, XBOOLE_0:def 10;
  hence A8: v1.inDegree() = v2.inDegree();
  now
    let e be object;
    assume e in v1.edgesOut();
    then consider x being set such that
      A9: e DJoins v1,x,G by GLIB_000:59;
    set W = G.walkOf(v1,e,x);
    A10: e Joins v1,x,G by A9, GLIB_000:16;
    then W is_Walk_from v1,x by GLIB_001:15;
    then A11: x in G.reachableFrom(v1) by GLIB_002:def 5;
    A12: e Joins v2,x,G & e DJoins v2,x,G by A1, A9, A10;
    the_Vertices_of C = G.reachableFrom(v1) by A1, GLIB_002:33;
    then e in G.edgesBetween(the_Vertices_of C) by A11, A12, GLIB_000:32;
    then e in the_Edges_of C by GLIB_002:31;
    then e DJoins v2,x,C & e is set by A12, GLIB_000:73;
    hence e in v2.edgesOut() by GLIB_000:59;
  end;
  then v1.edgesOut() c= v2.edgesOut() by TARSKI:def 3;
  hence A13: v1.edgesOut() = v2.edgesOut() by A2, XBOOLE_0:def 10;
  hence A14: v1.outDegree() = v2.outDegree();
  thus v1.edgesInOut() = v2.edgesInOut() by A7, A13;
  thus v1.degree() = v2.degree() by A8, A14;
end;
