
theorem
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is onto semi-Dcontinuous holds G2.supDegree() c= G1.supDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is onto semi-Dcontinuous;
  set D1 = the set of all v.degree() where v is Vertex of G1;
  set D2 = the set of all w.degree() where w is Vertex of G2;
  now
    let x be object;
    assume x in G2.supDegree();
    then consider d2 being set such that
      A2: x in d2 & d2 in D2 by TARSKI:def 4;
    consider w being Vertex of G2 such that
      A3: d2 = w.degree() by A2;
    rng F_V = the_Vertices_of G2 by A1, GLIB_010:def 12;
    then consider v being object such that
      A4: v in dom F_V & F_V.v = w by FUNCT_1:def 3;
    reconsider v as Vertex of G1 by A4;
    (F_V/.v).degree() c= v.degree() by A1, A4, GLIBPRE0:91;
    then w.degree() c= v.degree() by A4, PARTFUN1:def 6;
    then A5: x in v.degree() by A2, A3;
    v.degree() in D1;
    hence x in G1.supDegree() by A5, TARSKI:def 4;
  end;
  hence thesis by TARSKI:def 3;
end;
