
theorem Th59:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is onto semi-Dcontinuous & dom F_V = the_Vertices_of G1 holds
    G2.minInDegree() c= G1.minInDegree() &
    G2.minOutDegree() c= G1.minOutDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is onto semi-Dcontinuous & dom F_V = the_Vertices_of G1;
  consider v1 being Vertex of G1 such that
    A2: v1.inDegree() = G1.minInDegree() and
    for w1 being Vertex of G1 holds v1.inDegree() c= w1.inDegree() by Th37;
  consider v2 being Vertex of G2 such that
    A3: v2.inDegree() = G2.minInDegree() and
    A4: for w2 being Vertex of G2 holds v2.inDegree() c= w2.inDegree() by Th37;
  A5: (F_V/.v1).inDegree() c= v1.inDegree() by A1, GLIBPRE0:90;
  v2.inDegree() c= (F_V/.v1).inDegree() by A4;
  hence G2.minInDegree() c= G1.minInDegree() by A2, A3, A5, XBOOLE_1:1;
  consider v3 being Vertex of G1 such that
    A6: v3.outDegree() = G1.minOutDegree() and
    for w3 being Vertex of G1 holds v3.outDegree() c= w3.outDegree() by Th38;
  consider v4 being Vertex of G2 such that
    A7: v4.outDegree() = G2.minOutDegree() and
    A8: for w4 being Vertex of G2 holds v4.outDegree()c=w4.outDegree() by Th38;
  A9: (F_V/.v3).outDegree() c= v3.outDegree() by A1, GLIBPRE0:90;
  v4.outDegree() c= (F_V/.v3).outDegree() by A8;
  hence G2.minOutDegree() c= G1.minOutDegree() by A6, A7, A9, XBOOLE_1:1;
end;
