
theorem Th57:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is directed weak_SG-embedding & rng F_V = the_Vertices_of G2 holds
    G1.minInDegree() c= G2.minInDegree() &
    G1.minOutDegree() c= G2.minOutDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is directed weak_SG-embedding & rng F_V = the_Vertices_of G2;
  consider v1 being Vertex of G1 such that
    A2: v1.inDegree() = G1.minInDegree() and
    A3: for w1 being Vertex of G1 holds v1.inDegree() c= w1.inDegree() by Th37;
  consider v2 being Vertex of G2 such that
    A4: v2.inDegree() = G2.minInDegree() and
    for w2 being Vertex of G2 holds v2.inDegree() c= w2.inDegree() by Th37;
  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.inDegree() c= (F_V/.v0).inDegree() by A1, GLIBPRE0:88;
  then A6: v0.inDegree() c= v2.inDegree() by A5, PARTFUN1:def 6;
  v1.inDegree() c= v0.inDegree() by A3;
  hence G1.minInDegree() c= G2.minInDegree() by A2, A4, A6, XBOOLE_1:1;
  consider v3 being Vertex of G1 such that
    A7: v3.outDegree() = G1.minOutDegree() and
    A8: for w3 being Vertex of G1 holds v3.outDegree()c=w3.outDegree() by Th38;
  consider v4 being Vertex of G2 such that
    A9: v4.outDegree() = G2.minOutDegree() and
    for w4 being Vertex of G2 holds v4.outDegree() c= w4.outDegree() by Th38;
  consider v9 being object such that
    A10: v9 in dom F_V & F_V.v9 = v4 by A1, FUNCT_1:def 3;
  reconsider v9 as Vertex of G1 by A10;
  v9.outDegree() c= (F_V/.v9).outDegree() by A1, GLIBPRE0:88;
  then A11: v9.outDegree() c= v4.outDegree() by A10, PARTFUN1:def 6;
  v3.outDegree() c= v9.outDegree() by A8;
  hence thesis by A7, A9, A11, XBOOLE_1:1;
end;
