
theorem Th52:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is weak_SG-embedding & rng F_V = the_Vertices_of G2 holds
    G1.minDegree() c= G2.minDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding & rng F_V = the_Vertices_of G2;
  consider v1 being Vertex of G1 such that
    A2: v1.degree() = G1.minDegree() and
    A3: for w1 being Vertex of G1 holds v1.degree() c= w1.degree() by Th36;
  consider v2 being Vertex of G2 such that
    A4: v2.degree() = G2.minDegree() and
    for w2 being Vertex of G2 holds v2.degree() c= w2.degree() by Th36;
  consider v0 being object such that
    A5: v0 in dom F_V & F_V.v0 = v2 by A1, FUNCT_1:def 3;
  reconsider v0 as Vertex of G1 by A5;
  v0.degree() c= (F_V/.v0).degree() by A1, GLIBPRE0:89;
  then A6: v0.degree() c= v2.degree() by A5, PARTFUN1:def 6;
  v1.degree() c= v0.degree() by A3;
  hence thesis by A2, A4, A6, XBOOLE_1:1;
end;
