
theorem Th50:
  for G being _Graph, c being Cardinal
  st ex v being Vertex of G st v.outDegree() = c &
    for w being Vertex of G holds w.outDegree() c= v.outDegree()
  holds G.supOutDegree() = c
proof
  let G be _Graph, c be Cardinal;
  given v being Vertex of G such that
    A1: v.outDegree() = c and
    A2: for w being Vertex of G holds w.outDegree() c= v.outDegree();
  set S = the set of all v.outDegree() where v is Vertex of G;
  c in S by A1;
  then A3: c c= G.supOutDegree() by ZFMISC_1:74;
  now
    let x be object;
    assume x in G.supOutDegree();
    then consider X being set such that
      A4: x in X & X in S by TARSKI:def 4;
    consider w being Vertex of G such that
      A5: X = w.outDegree() by A4;
    X c= v.outDegree() by A2, A5;
    hence x in c by A1, A4;
  end;
  then G.supOutDegree() c= c by TARSKI:def 3;
  hence thesis by A3, XBOOLE_0:def 10;
end;
