
theorem
  for c being Cardinal, G being _trivial c-edge _Graph holds
    G.maxInDegree() = c & G.minInDegree() = c &
    G.maxOutDegree() = c & G.minOutDegree() = c &
    G.maxDegree() = c +` c & G.minDegree() = c +` c
proof
  let c be Cardinal, G be _trivial c-edge _Graph;
  set v = the Vertex of G;
  now
    let w be Vertex of G;
    w.inDegree() = G.size() & v.inDegree() = G.size() by GLIB_000:149;
    hence w.inDegree() c= v.inDegree();
  end;
  hence G.maxInDegree() = G.size() by Th49, GLIB_000:149
    .= c by Def4;
  now
    let w be Vertex of G;
    w.inDegree() = G.size() & v.inDegree() = G.size() by GLIB_000:149;
    hence v.inDegree() c= w.inDegree();
  end;
  hence G.minInDegree() = G.size() by Th37, GLIB_000:149
    .= c by Def4;
  now
    let w be Vertex of G;
    w.outDegree() = G.size() & v.outDegree() = G.size() by GLIB_000:149;
    hence w.outDegree() c= v.outDegree();
  end;
  hence G.maxOutDegree() = G.size() by Th50, GLIB_000:149
    .= c by Def4;
  now
    let w be Vertex of G;
    w.outDegree() = G.size() & v.outDegree() = G.size() by GLIB_000:149;
    hence v.outDegree() c= w.outDegree();
  end;
  hence G.minOutDegree() = G.size() by Th38, GLIB_000:149
    .= c by Def4;
  now
    let w be Vertex of G;
    w.degree() = G.size()+`G.size() & v.degree() = G.size()+`G.size()
      by GLIB_000:149;
    hence w.degree() c= v.degree();
  end;
  hence G.maxDegree() = G.size()+`G.size() by Th48, GLIB_000:149
    .= G.size() +` c by Def4
    .= c +` c by Def4;
  now
    let w be Vertex of G;
    w.degree() = G.size()+`G.size() & v.degree() = G.size()+`G.size()
      by GLIB_000:149;
    hence v.degree() c= w.degree();
  end;
  hence G.minDegree() = G.size()+`G.size() by Th36, GLIB_000:149
    .= G.size() +` c by Def4
    .= c +` c by Def4;
end;
