
theorem Th67:
for G being SimpleGraph, C being Coloring of G, S being Subset of Vertices G
  holds C | S is Coloring of (G SubgraphInducedBy S)
proof
 let G be SimpleGraph, C be Coloring of G, S be Subset of Vertices G;
  set g = G SubgraphInducedBy S;
A1: Vertices g = S /\ Vertices G by Th45 .= S by XBOOLE_1:28;
  reconsider CS = C | S as a_partition of Vertices g by A1;
  CS is StableSet-wise proof
    let x be set such that
  A2: x in CS;
      reconsider xx = x as Subset of Vertices g by A2;
      consider z being Element of C such that
  A3: xx = z /\ S and z meets S by A2;
      xx is stable proof
        let x, y be set such that
      A4: x <> y and
      A5: x in xx and
      A6: y in xx;
        assume A7: {x,y} in g;
      A8: x in z by A3,A5,XBOOLE_0:def 4;
      A9: y in z by A6,A3,XBOOLE_0:def 4;
          z is StableSet of G by A3,A5,Def20;
        hence contradiction by A7,A4,A8,A9,Def19;
      end;
    hence x is StableSet of g;
  end;
 hence C | S is Coloring of g;
end;
