
theorem Th38:
  for G being _Graph, c being Cardinal holds G.minOutDegree() = c iff
    ex v being Vertex of G st v.outDegree() = c &
      for w being Vertex of G holds v.outDegree() c= w.outDegree()
proof
  let G be _Graph, c be Cardinal;
  set S = the set of all v.outDegree() where v is Vertex of G;
  (the Vertex of G).outDegree() in S;
  then A1: S <> {};
  now
    let x be set;
    assume x in S;
    then consider v being Vertex of G such that
      A2: x = v.outDegree();
    thus x is cardinal number by A2;
  end;
  then consider A being Cardinal such that
    A3: A in S & A = G.minOutDegree() by A1, GLIBPRE0:14;
  hereby
    assume A4: G.minOutDegree() = c;
    consider v being Vertex of G such that
      A5: A = v.outDegree() by A3;
    take v;
    thus v.outDegree() = c by A4, A3, A5;
    let w be Vertex of G;
    w.outDegree() in S;
    hence v.outDegree() c= w.outDegree() by A3, A5, SETFAM_1:3;
  end;
  given v being Vertex of G such that
    A6: v.outDegree() = c and
    A7: for w being Vertex of G holds v.outDegree() c= w.outDegree();
  c in S by A6;
  then A8: G.minOutDegree() c= c by SETFAM_1:3;
  now
    let x be object;
    assume A9: x in c;
    now
      let X be set;
      assume X in S;
      then consider w being Vertex of G such that
        A10: X = w.outDegree();
      c c= X by A6, A7, A10;
      hence x in X by A9;
    end;
    hence x in meet S by A1, SETFAM_1:def 1;
  end;
  then c c= G.minOutDegree() by TARSKI:def 3;
  hence G.minOutDegree() = c by A8, XBOOLE_0:def 10;
end;
