
theorem Th80:
  for G being _Graph st G is with_max_in_degree
  holds ex v being Vertex of G st v.inDegree() = G.supInDegree() &
    for w being Vertex of G holds w.inDegree() c= v.inDegree()
proof
  let G be _Graph;
  assume G is with_max_in_degree;
  then consider v being Vertex of G such that
    A1: for w being Vertex of G holds w.inDegree() c= v.inDegree();
  take v;
  set D = the set of all w.inDegree() where w is Vertex of G;
  now
    let X be set;
    assume X in D;
    then consider w being Vertex of G such that
      A2: X = w.inDegree();
    thus X c= v.inDegree() by A1, A2;
  end;
  then A3: union D c= v.inDegree() by ZFMISC_1:76;
  v.inDegree() c= G.supInDegree() by Th35;
  hence thesis by A1, A3, XBOOLE_0:def 10;
end;
