
theorem Th56:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is directed weak_SG-embedding holds
    G1.supInDegree() c= G2.supInDegree() &
    G1.supOutDegree() c= G2.supOutDegree()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is directed weak_SG-embedding;
  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 G1.supInDegree();
    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.inDegree() by A2;
    v.inDegree() c= (F_V/.v).inDegree() by A1, GLIBPRE0:88;
    then A4: x in (F_V/.v).inDegree() by A2, A3;
    (F_V/.v).inDegree() in D2;
    hence x in G2.supInDegree() by A4, TARSKI:def 4;
  end;
  hence G1.supInDegree() c= G2.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 G1.supOutDegree();
    then consider d1 being set such that
      A5: x in d1 & d1 in D3 by TARSKI:def 4;
    consider v being Vertex of G1 such that
      A6: d1 = v.outDegree() by A5;
    v.outDegree() c= (F_V/.v).outDegree() by A1, GLIBPRE0:88;
    then A7: x in (F_V/.v).outDegree() by A5, A6;
    (F_V/.v).outDegree() in D4;
    hence x in G2.supOutDegree() by A7, TARSKI:def 4;
  end;
  hence G1.supOutDegree() c= G2.supOutDegree() by TARSKI:def 3;
end;
