
theorem Th58:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is onto semi-Dcontinuous holds
    G2.supInDegree() c= G1.supInDegree() &
    G2.supOutDegree() c= G1.supOutDegree()
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.inDegree() where v is Vertex of G1;
  set D2 = the set of all w.inDegree() where w is Vertex of G2;
  now
    let x be object;
    assume x in G2.supInDegree();
    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.inDegree() 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).inDegree() c= v.inDegree() by A1, A4, GLIBPRE0:90;
    then w.inDegree() c= v.inDegree() by A4, PARTFUN1:def 6;
    then A5: x in v.inDegree() by A2, A3;
    v.inDegree() in D1;
    hence x in G1.supInDegree() by A5, TARSKI:def 4;
  end;
  hence G2.supInDegree() c= G1.supInDegree() by TARSKI:def 3;
  set D3 = the set of all v.outDegree() where v is Vertex of G1;
  set D4 = the set of all w.outDegree() where w is Vertex of G2;
  now
    let x be object;
    assume x in G2.supOutDegree();
    then consider d2 being set such that
      A6: x in d2 & d2 in D4 by TARSKI:def 4;
    consider w being Vertex of G2 such that
      A7: d2 = w.outDegree() by A6;
    rng F_V = the_Vertices_of G2 by A1, GLIB_010:def 12;
    then consider v being object such that
      A8: v in dom F_V & F_V.v = w by FUNCT_1:def 3;
    reconsider v as Vertex of G1 by A8;
    (F_V/.v).outDegree() c= v.outDegree() by A1, A8, GLIBPRE0:90;
    then w.outDegree() c= v.outDegree() by A8, PARTFUN1:def 6;
    then A9: x in v.outDegree() by A6, A7;
    v.outDegree() in D3;
    hence x in G1.supOutDegree() by A9, TARSKI:def 4;
  end;
  hence G2.supOutDegree() c= G1.supOutDegree() by TARSKI:def 3;
end;
