
theorem Th81:
  for G being _Graph st G is with_max_out_degree
  holds ex v being Vertex of G st v.outDegree() = G.supOutDegree() &
    for w being Vertex of G holds w.outDegree() c= v.outDegree()
proof
  let G be _Graph;
  assume G is with_max_out_degree;
  then consider v being Vertex of G such that
    A1: for w being Vertex of G holds w.outDegree() c= v.outDegree();
  take v;
  set D = the set of all w.outDegree() 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.outDegree();
    thus X c= v.outDegree() by A1, A2;
  end;
  then A3: union D c= v.outDegree() by ZFMISC_1:76;
  v.outDegree() c= G.supOutDegree() by Th35;
  hence thesis by A1, A3, XBOOLE_0:def 10;
end;
