
theorem Th46:
  for G being _Graph st ex v being Vertex of G st v is isolated
  holds G.minDegree() = 0 & G.minInDegree() = 0 & G.minOutDegree() = 0
proof
  let G be _Graph;
  given v being Vertex of G such that
    A1: v is isolated;
  A2: v.degree() = 0 by A1, GLIB_000:157;
  for w be Vertex of G holds v.degree() c= w.degree() by A2, XBOOLE_1:2;
  hence G.minDegree() = 0 by A2, Th36;
  A3: v.inDegree() = 0 & v.outDegree() = 0 by A1, GLIB_000:156;
  for w be Vertex of G holds v.inDegree() c= w.inDegree() by A3, XBOOLE_1:2;
  hence G.minInDegree() = 0 by A3, Th37;
  for w be Vertex of G holds v.outDegree() c= w.outDegree() by A3, XBOOLE_1:2;
  hence G.minOutDegree() = 0 by A3, Th38;
end;
