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

theorem Th61:
  for c being Cardinal, G being _trivial c-edge _Graph, v being Vertex of G
  holds G.inDegreeMap() = v .--> c & G.outDegreeMap() = v .--> c &
    G.degreeMap() = v .--> (2*`c)
proof
  let c be Cardinal, G be _trivial c-edge _Graph, v be Vertex of G;
  consider v0 being Vertex of G such that
    A1: the_Vertices_of G = {v0} by GLIB_000:22;
  set f = v .--> c;
  A2: v = v0 by A1, TARSKI:def 1;
  dom f = dom({v}-->c) by FUNCOP_1:def 9
    .= the_Vertices_of G by A1, A2;
  then A3: dom f = dom G.inDegreeMap() & dom f = dom G.outDegreeMap()
    by PARTFUN1:def 2;
  now
    let x be object;
    assume x in dom G.inDegreeMap();
    then reconsider w = x as Vertex of G;
    A4: w = v by A1, A2, TARSKI:def 1;
    hence f.x = c by FUNCOP_1:72
      .= G.size() by GLIB_013:def 4
      .= v.inDegree() by GLIB_000:149
      .= G.inDegreeMap().x by A4, Def12;
  end;
  hence G.inDegreeMap() = f by A3, FUNCT_1:2;
  now
    let x be object;
    assume x in dom G.outDegreeMap();
    then reconsider w = x as Vertex of G;
    A5: w = v by A1, A2, TARSKI:def 1;
    hence f.x = c by FUNCOP_1:72
      .= G.size() by GLIB_013:def 4
      .= v.outDegree() by GLIB_000:149
      .= G.outDegreeMap().x by A5, Def13;
  end;
  hence G.outDegreeMap() = f by A3, FUNCT_1:2;
  set g = v .--> (2*`c);
  A6: dom g = dom({v}-->2*`c) by FUNCOP_1:def 9
    .= dom G.degreeMap() by A1, A2, PARTFUN1:def 2;
  now
    let x be object;
    assume x in dom G.degreeMap();
    then reconsider w = x as Vertex of G;
    A7: w = v by A1, A2, TARSKI:def 1;
    hence g.x = 2*`c by FUNCOP_1:72
      .= c +` c by CARD_2:23
      .= G.size() +` c by GLIB_013:def 4
      .= G.size() +` G.size() by GLIB_013:def 4
      .= v.degree() by GLIB_000:149
      .= G.degreeMap().x by A7, Def11;
  end;
  hence thesis by A6, FUNCT_1:2;
end;
