reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem Th44:
  for G being c-Dregular _Graph holds
    G.minInDegree() = c & G.minOutDegree() = c &
    G.maxInDegree() = c & G.maxOutDegree() = c
proof
  let G be c-Dregular _Graph;
  now
    set v = the Vertex of G;
    take v;
    thus A1: v.inDegree() = c by Def8;
    let w be Vertex of G;
    thus v.inDegree() c= w.inDegree() by A1, Def8;
  end;
  hence G.minInDegree() = c by GLIB_013:37;
  now
    set v = the Vertex of G;
    take v;
    thus A2: v.outDegree() = c by Def8;
    let w be Vertex of G;
    thus v.outDegree() c= w.outDegree() by A2, Def8;
  end;
  hence G.minOutDegree() = c by GLIB_013:38;
  now
    set v = the Vertex of G;
    take v;
    thus A3: v.inDegree() = c by Def8;
    let w be Vertex of G;
    thus w.inDegree() c= v.inDegree() by A3, Def8;
  end;
  hence G.maxInDegree() = c by GLIB_013:49;
  now
    set v = the Vertex of G;
    take v;
    thus A4: v.outDegree() = c by Def8;
    let w be Vertex of G;
    thus w.outDegree() c= v.outDegree() by A4, Def8;
  end;
  hence G.maxOutDegree() = c by GLIB_013:50;
end;
