
theorem
  for G being locally-finite _Graph, n being Nat
  holds G.minInDegree() = n iff
    ex v being Vertex of G st v.inDegree() = n &
      for w being Vertex of G holds v.inDegree() <= w.inDegree()
proof
  let G be locally-finite _Graph, n be Nat;
  hereby
    assume G.minInDegree() = n;
    then consider v being Vertex of G such that
      A1: v.inDegree() = n and
      A2: for w being Vertex of G holds v.inDegree() c= w.inDegree() by Th37;
    take v;
    thus v.inDegree() = n by A1;
    let w be Vertex of G;
    Segm v.inDegree() c= Segm w.inDegree() by A2;
    hence v.inDegree() <= w.inDegree() by NAT_1:39;
  end;
  given v being Vertex of G such that
    A3: v.inDegree() = n and
    A4: for w being Vertex of G holds v.inDegree() <= w.inDegree();
  now
    let w be Vertex of G;
    Segm v.inDegree() c= Segm w.inDegree() by A4, NAT_1:39;
    hence v.inDegree() c= w.inDegree();
  end;
  hence thesis by A3, Th37;
end;
