
theorem Th68:
for G, H being finitely_colorable SimpleGraph
 st G c= H holds chromatic# G <= chromatic# H
proof
 let G, H be finitely_colorable SimpleGraph;
 assume A1: G c= H;
  then reconsider S = Vertices G as Subset of Vertices H by ZFMISC_1:77;
  set g = H SubgraphInducedBy S;
A2: G c= g by A1,Th44;
  consider C being finite Coloring of H such that
A3: card C = chromatic# H by Def22;
  reconsider g as finitely_colorable SimpleGraph;
  reconsider Cg = C | S as finite Coloring of g by Th67;
A4: Vertices G = Vertices g by Lm9;
A5: G c= g proof
      let a be object;
      assume a in G;
       then a in { {} } \/ singletons Vertices G \/ Edges G by Th27;
       then A6: a in { {} } \/ singletons Vertices G or a in Edges G
            by XBOOLE_0:def 3;
       per cases by A6,XBOOLE_0:def 3;
       suppose a in {{}};
          then a = {} by TARSKI:def 1;
         hence a in g by Th20;
       end;
       suppose a in singletons Vertices G;
          then a in {{}} \/ singletons Vertices g by A4,XBOOLE_0:def 3;
          then a in {{}} \/ singletons Vertices g \/ Edges g by XBOOLE_0:def 3;
         hence a in g by Th27;
       end;
       suppose a in Edges G;
           then a in G;
         hence a in g by A2;
       end;
    end;
   reconsider Cg1 = Cg as a_partition of Vertices G;
   Cg1 is StableSet-wise proof
     let x be set such that
   A7: x in Cg1;
       reconsider xx = x as Subset of Vertices G by A7;
       reconsider xxx = x as Subset of Vertices g by A7;
     xx is stable proof
      let x, y be set such that
     A8: x <> y and
     A9: x in xx and
     A10: y in xx;
     A11: xxx is stable by A7,Def20;
       assume {x,y} in G;
       hence contradiction by A5,A8,A9,A10,A11;
     end;
     hence x is StableSet of G;
   end; then
A12: card Cg >= chromatic# G by Def22;
   card C >= card (C | S) by MYCIELSK:8;
 hence chromatic# G <= chromatic# H by A12,A3,XXREAL_0:2;
end;
