
theorem Th51:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is weak_SG-embedding holds G1.supDegree() c= G2.supDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding;
  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 G1.supDegree();
    then consider d1 being set such that
      A2: x in d1 & d1 in D1 by TARSKI:def 4;
    consider v being Vertex of G1 such that
      A3: d1 = v.degree() by A2;
    v.degree() c= (F_V/.v).degree() by A1, GLIBPRE0:89;
    then A4: x in (F_V/.v).degree() by A2, A3;
    (F_V/.v).degree() in D2;
    hence x in G2.supDegree() by A4, TARSKI:def 4;
  end;
  hence G1.supDegree() c= G2.supDegree() by TARSKI:def 3;
end;
