reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;

theorem
  for G1 being _Graph, G2 being loopless _Graph
  for F being PGraphMapping of G1, G2 st F is weak_SG-embedding
  holds G1.vChromaticNum() c= G2.vChromaticNum()
proof
  let G1 be _Graph, G2 be loopless _Graph, F be PGraphMapping of G1, G2;
  assume F is weak_SG-embedding;
  then G1 is G2.vChromaticNum()-vcolorable by Th42, Th54;
  hence thesis by Th57;
end;
