reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem Th64:
  for G being _finite _Graph, v being Denumeration of the_Vertices_of G
  holds G.degreeMap()*v = G.inDegreeMap()*v + G.outDegreeMap()*v
proof
  let G be _finite _Graph, v be Denumeration of the_Vertices_of G;
  dom G.degreeMap() = the_Vertices_of G by PARTFUN1:def 2
    .= rng v by FUNCT_2:def 3;
  then A1: dom(G.degreeMap()*v) = dom v by RELAT_1:27;
  dom G.inDegreeMap() = the_Vertices_of G by PARTFUN1:def 2
    .= rng v by FUNCT_2:def 3;
  then A2: dom(G.inDegreeMap()*v) = dom v by RELAT_1:27;
  dom G.outDegreeMap() = the_Vertices_of G by PARTFUN1:def 2
    .= rng v by FUNCT_2:def 3;
  then dom(G.outDegreeMap()*v) = dom v by RELAT_1:27;
  then A3: dom(G.degreeMap()*v)=dom(G.inDegreeMap()*v)/\dom(G.outDegreeMap()*v)
    by A1, A2;
  now
    let c be object;
    assume c in dom(G.degreeMap()*v);
    then A4: c in dom v by A1;
    then v.c in rng v by FUNCT_1:3;
    then reconsider w = v.c as Vertex of G;
    thus (G.degreeMap()*v).c = G.degreeMap().w by A4, FUNCT_1:13
      .= G.inDegreeMap().w +` G.outDegreeMap().w by Th60
      .= (G.inDegreeMap()*v).c + G.outDegreeMap().w by A4, FUNCT_1:13
      .= (G.inDegreeMap()*v).c + (G.outDegreeMap()*v).c by A4, FUNCT_1:13;
  end;
  hence thesis by A3, VALUED_1:def 1;
end;
